Donor Umbilical Cord Blood Transplant in Treating Patients With Hematologic Cancer

Inclusion Criteria: * Patients \> 70 may be considered if performance status \> 80% or Eastern Cooperative Oncology Group (ECOG) =\< 1 and comorbidity score \< 3; these patients must be discussed with the principal investigator (PI), Rachel Salit prior to enrollment * Adequate cardiac function defined as absence of decompensated congestive heart failure, or uncontrolled arrhythmia and: * Left ventricular ejection fraction \>= 35% or * Fractional shortening \> 22% * Adequate pulmonary function defined as diffusion capacity of carbon monoxide (DLCO) \> 30% predicted, and absence of oxygen (O2) requirements * Adequate hepatic function; patients with clinical or laboratory evidence of liver disease will be evaluated for the cause of liver disease, its clinical severity in terms of liver function, histology, and the degree of portal hypertension; patients with fulminant liver failure, cirrhosis with evidence of portal hypertension or bridging fibrosis, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, or correctable hepatic synthetic dysfunction evidenced by prolongation of the prothrombin time, ascites related to portal hypertension, bacterial or fungal abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \> 3 mg/dL, and symptomatic biliary disease will be excluded * Adequate renal function defined as creatinine =\< 2.0 mg/dl (adults) or creatinine clearance \> 40 ml/min (pediatrics) * All adults with a creatinine \> 1.2 or a history of renal dysfunction must have estimated creatinine clearance \> 40 ml/min * Performance status score: Karnofsky (for adults) \>= 60 or ECOG 0-2; Lansky (for children) score \>= 50 * If recent mold infection, e.g., Aspergillus, must be cleared by infectious disease * Second hematopoietic cell transplant: Must be \>= 3 months after prior myeloablative transplant * Patients who have received \< 2 cycles of multiagent chemotherapy and patients who have received no multiagent chemotherapy within the 3 months previous to umbilical cord blood transplant (UCBT) as well as patients experiencing graft failure following previous allogeneic transplant * Acute myeloid leukemia/acute lymphoblastic leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia: Must have \< 5% morphologic marrow blasts in an evaluable marrow (\> 25% of normal cellularity for age) collected less than one month prior to start of conditioning; patients persistently aplastic for greater than one month since completing last chemotherapy are also eligible with the approval of the PI or designee * Chronic myelogenous leukemia: All types, except refractory blast crisis; chronic phase patients must have failed or been intolerant to Gleevec or other tyrosine kinase inhibitors; at time of transplant, patients must have \< 5% blasts in an evaluable marrow (\> 25% of normal cellularity for age) by morphology within the bone marrow * Myelodysplastic syndrome (MDS): Any subtype; morphologic blasts must be less than 5% in an evaluable marrow (\> 25% of normal cellularity for age); if blasts are 5% or more, patient requires induction chemotherapy pre-transplant to reduce blast count to less than 5%; patients who have a hypocellular marrow in the absence of excess blasts that is related to the underlying disease or as a result of treatment for MDS may also be eligible with the approval of the PI or designee * Large-cell lymphoma and aggressive T-cell lymphoma: With chemotherapy sensitive disease that has failed autologous transplant or patients who are ineligible for an autologous transplant; chemotherapy sensitive disease is defined as \>= 50% reduction in the size of the tumor with the chemotherapy regimen immediately preceding transplant * Chronic lymphocytic leukemia/small lymphocytic lymphoma (CLL/SLL): Must be refractory to fludarabine (fludarabine phosphate) or fail to have a complete or partial response after therapy with a regimen containing fludarabine (or another nucleoside analog, e.g. cladribine \[2-CDA\], pentostatin) or experience disease relapse within 12 months after completing therapy with a regimen containing fludarabine (or another nucleoside analog) * Hodgkin disease: Must have received and failed frontline therapy * Follicular lymphoma, marginal zone B-cell lymphoma, lymphoplasmacytic lymphoma, mantle-cell lymphoma, and indolent T-cell lymphomas: Must have progressed with the most recent remission duration being \< 6 months; patients with bulky disease should be considered for debulking chemotherapy before transplant; patients with refractory disease are eligible, unless they have bulky disease and an estimated tumor doubling time of less than one month * Multiple myeloma: Must have received prior chemotherapy; consolidation of chemotherapy by autografting prior to nonmyeloablative hematopoietic cell transplant (HCT) is permitted * Myeloproliferative syndromes * DONOR: Cord blood (CB) donor selection will be based on institutional guidelines and in general should be selected to optimize both human leukocyte antigen (HLA) match and cell dose; additionally, CB grafts shall consist of one or two CB donors based on, but not exclusively determined by, cell dose (total nucleated cell \[TNC\]/kg and CD34/kg), HLA matching and disease status and indication for transplant; attending preference will be allowed for single versus double unit as well as the degree of mismatching based on patient specific factors, as long as the following minimum criteria are met: * HLA matching * Minimum requirement: The CB graft(s) must be matched at a minimum at 4/6 HLA-A, B, DRB1 loci with the recipient. Therefore 0-2 mismatches at the A or B or DRB1 loci based on intermediate resolution A, B antigen and DRB1 allele typing for determination of HLA-match is allowed * HLA-matching determined by high-resolution typing is allowed per institutional guidelines as long as the minimum criteria are met * Selection of two CB units is mandatory when a single cord blood unit does not meet the following criteria: * Match grade 6/6; TNC Dose \>= 2.5 x 10\^7/kg * Match grade 5/6 or 4/6; TNC dose \>= 4.0 (+/- 0.5) x 10\^7/kg * If two CB units are used, the total cell dose of the combined units must be at least 3.0 x 10\^7 TNC per kilogram recipient weight based on pre-cryopreservation numbers, with each CB unit containing a MINIMUM of 1.5 x 10\^7 TNC/kg * The minimum recommended CD34/kg cell dose should be 2 x 10\^5 CD34/kg, total dose from a single or combined double * The unmanipulated CB unit(s) will be Food and Drug Administration (FDA) licensed or will be obtained under a separate investigational new drug (IND), such as the National Marrow Donor Program (NMDP) Protocol 10-CBA conducted under BB IND-7555 or another IND sponsored by (1) a participating institution or (2) an investigator at FHCRC or one of the participating institutions * FHCRC only: Up to 5% of cord blood product, when ready for infusion, may be withheld for research purposes as long as thresholds for infused TNC dose are met; threshold for double unit transplantation is \>= 3.0 x 10\^7/kg; these products will be used to conduct studies involving the immunobiology of double cord transplantation and kinetics of engraftment Exclusion Criteria: * Patients with an available 5-6/6 HLA-A, B, DRB1 matched sibling donor * Pregnancy or breastfeeding * Evidence of human immunodeficiency virus (HIV) infection or known HIV positive serology * Uncontrolled viral or bacterial infection at the time of study enrollment * Active or recent (prior 6 month) invasive fungal infection without infectious disease (ID) consult and approval * Active central nervous system malignancy * Patients who have received \>= 2 cycles of multiagent chemotherapy within the 3 months previous to UCBT; patients who have had previous autologous transplant within 12 months of UCBT are excluded regardless of history of recent treatment * DONOR: Any cord blood units with \< 1.5 x 10\^7 total nucleated cells per kilogram recipient weight * DONOR: Any cord blood units without the full maternal testing and negative results for hepatitis A, B, C, HIV, and human T-lymphotropic virus (HTLV-1) viruses; any additional available virology results on the unit itself will be reviewed but are not mandated, complete or always available; cord blood units are presumed to be cytomegalovirus (CMV) negative regardless of serologic testing due to passive transmission of maternal CMV antibodies