Selective Depletion of CD45RA+ T Cells From Allogeneic Peripheral Blood Stem Cell Grafts From HLA-Matched Related and Unrelated Donors in Preventing GVHD

Inclusion Criteria: * Patients who are considered appropriate candidates for allogeneic hematopoietic stem cell transplantation and have one of the following diagnoses: * Acute lymphocytic leukemia in first or subsequent remission * Acute myeloid leukemia in first or subsequent remission * Acute lymphocytic leukemia in relapse or primary refractory disease with a circulating blast count of no more than 10,000/mm\^3 (Arms A or C only) * Acute myeloid leukemia in relapse or primary refractory disease with a circulating blast count of no more than 10,000/mm\^3 (Arms A or C only) * Refractory anemia with excess blasts (RAEB-1 and RAEB-2) (if the patient has received previous induction chemotherapy within 60 days) * Chronic myelogenous leukemia with a history of accelerated phase or blast crisis (if the patient has received at least one course of induction chemotherapy) * Other acute leukemia or related neoplasm (including but not limited to 'biphenotypic', 'undifferentiated' or 'ambiguous lineage' acute leukemia, blastic plasmacytoid dendritic cell neoplasm or lymphoblastic lymphoma) * Patients 0-49 years old will be enrolled in Arm A or C (high-intensity) * Patients 50-60 years old will be enrolled in Arm B or D (lower intensity); patients eligible for Arms B or D also include those who have received previous allogeneic HCT, or who have co-morbid conditions rendering them unsuitable for high-dose conditioning, determined in consultation with the principal investigator * Patient with a HLA-matched (HLA-A, B, C, and DR beta 1 \[DRB1\] molecularly matched) unrelated donor or related donor capable of donating PBSC * DONOR INCLUSION: * HLA-matched related donors \>= 18 years and capable and willing to donate PBSC (Arms A and B) * HLA-matched unrelated donors (HLA-A, B, C, and DRB1 matched based on high-resolution typing) capable and willing to donate PBSC (Arms C and D) Exclusion Criteria: * Patients with central nervous system (CNS) involvement refractory to intrathecal chemotherapy and/or standard cranial-spinal radiation * Patients on other experimental protocols for prevention of acute GVHD * Patient weight \>= 100 kg; patients \>= 70 kg with MUDs must be discussed with the principal investigator * Patients who are human immunodeficiency virus positive (HIV+) * Patients with uncontrolled infections for whom myeloablative HCT is considered contraindicated by the consulting infectious disease physician (upper respiratory tract viral infection does not constitute an uncontrolled infection in this context) * Patients with organ dysfunction * ARM A OR C EXCLUSION: * Creatinine \> 1.5 mg/dl at the present time; patients with a known history of creatinine \> 1.5 mg/dl must have a current estimated creatinine clearance of \> 40 ml/min * Cardiac ejection fraction \< 45% * Diffusing capacity of the lung for carbon monoxide (DLCO) corrected \< 60%; patients who are unable to perform pulmonary function tests (for example, due to young age and/or developmental status) will be excluded if the oxygen (O2) saturation is \< 92% on room air * Liver function abnormality; patients who have liver function tests (LFTs) (including total bilirubin, aspartate aminotransferase \[AST\] and alanine aminotransferase \[ALT\]) \>= twice the upper limit of normal should be evaluated by a gastrointestinal (GI) physician; unless there is a clear precipitating factor (such as an azole, methotrexate, Bactrim or another drug); if the GI physician considers that HCT on the high-intensity arms of protocol is contraindicated for that patient the patient may be considered for treatment on the lower intensity arm of the protocol or excluded from the protocol; patients with Gilbert's syndrome and no other known liver function abnormality and patients with reversible drug-related transaminitis do not necessarily require GI consultation and may be included on the high-intensity arms of the protocol * ARM B OR D EXCLUSION: * Creatinine \> 2.0 mg/dl at the present time; patients with a known history of creatinine \> 1.5 mg/dl must have a current estimated creatinine clearance \> 40 ml/min * Cardiac ejection fraction \< 35% * DLCO corrected \< 50%; patients who are unable to perform pulmonary function tests (for example, due to young age and/or developmental status) will be excluded if the O2 saturation is \< 92% on room air; patients with a DLCO 50-60% must also have a partial pressure of oxygen (pO2) of \> 80 mmHg * Liver function abnormality; patients who have LFTs \>= twice the upper limit of normal should be evaluated by a GI physician unless there is a clear precipitating factor (such as an azole, methotrexate, Bactrim or another drug); 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 * Patients will be excluded from Arms A and C if they have received a previous myeloablative transplant; patients who have received a prior HCT at least 6 months prior may be considered for inclusion on Arms B or D after discussion with the principal investigator (PI) * Patients with a life expectancy \< 3 months from co-existing disease other than the leukemia or RAEB * Patients who are pregnant or breast-feeding * Fertile patients of child bearing age unwilling to use contraception during and for 12 months post-transplant * Patients with a significant other medical conditions that would make them unsuitable for transplant * Patients with a known hypersensitivity to tacrolimus, methotrexate (Arm A or C) or MMF (Arm B or D) * DONOR EXCLUSION: * Donors who are HIV-1, HIV-2, human T-lymphotropic virus (HTLV)-1, HTLV-2 seropositive or with active hepatitis B or hepatitis C virus infection * Donors who fail eligibility requirements for donation of cells or tissue for donation of a Human Cell and Tissue Products (HCT/P) will be excluded unless use of the cells complies urgent medical need or allogeneic use in a first-degree or second-degree relative * Unrelated donors donating outside of the United States of America (USA)