Reduced Intensity Chemotherapy and Radiation Therapy Before Donor Stem Cell Transplant in Treating Patients With Hematologic Malignancies

Inclusion Criteria: * Patients treated on this study will have: * Acute myeloid leukemia in morphologic complete remission (CR) not requiring treatment for their disease for 4 weeks * A history of acute myeloid leukemia (AML) with \< 10% residual blasts (use highest count on staging studies) after induction therapy and persisting with \< 10% blasts for at least 8 weeks without reinduction and at the time of HSCT * Refractory anemia (RA) or refractory anemia with ring sideroblasts (RARS) or isolated 5q- * Refractory anemia with excess blasts (RAEB)-1, refractory cytopenia with multilineage dysplasia (RCMD)+/-ringed sideroblasts (RS), or myelodysplastic syndrome (MDS) not otherwise specified (NOS) with stable disease for at least 3 months * RAEB-2 must demonstrate chemo-responsiveness; chemo-responsiveness is defined as a persistent blast percentage decrease by at least 5 percentage points to therapy and there must be =\< 10% blasts (use highest count on staging studies) after treatment and at the time of transplant * Hodgkin or Indolent non-Hodgkin's lymphoma * Myeloma with \< 5% plasma cells in the marrow * Myeloproliferative disorders (excludes chronic myelomonocytic leukemia \[CMML\]) * Aplastic anemia * A hematological or oncological disease (not listed) in which allogeneic HSCT is thought to be beneficial, and the disease is chemoresponsive * Patients without clear manifestation of their disease status in terms of stage and/or responsiveness should be discussed with the principal investigator (PI) and enrollment analysis should be documented in the study records * Patients must have a related donor who is human leukocyte antigen (HLA) mismatched at 2, 3, or 4 antigens at the HLA-A; B; C; DR loci in the graft-versus-host disease (GVHD) direction; (patients with related donors who are HLA identical or are a 1-antigen mismatch may be treated on this therapeutic approach, but will have their outcomes will not be part of the statistical aims of the study); the HLA matched related category includes patients with a syngeneic donor * Patients must have had front line therapy for their disease * LVEF (left ventricular end diastolic function) of \>= 45% * DLCO (diffusing capacity of the lung for carbon monoxide) \>= 45% of predicted corrected for hemoglobin, FEV-1 (forced expiratory volume at 1 second) \>= 50% of predicted * Serum bilirubin =\< 1.8 * Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) =\< 2.5 X upper limit of normal * Creatinine clearance of \>= 60 mL/min * HCT-CI/age score =\< 5 points (patients with greater than 5 points will be allowed for trial with approval of the PI and the co-PI or his designee; this is an adjustment to account for healthy patients who meet the spirit of this protocol but have histories that result in higher than HCT-CI 5 points; an example is a patient with a solid tumor malignancy in their remote history \[adds 3 points to HCT-CI total\] where the treatment for the malignancy occurred years to decades before and there has been complete recovery of toxicities) * Karnofsky performance status (KPS) \>= 90% patients older than 70 years, KPS \>= 80% patients younger than 70 years * Patients must be willing to use contraception if they have childbearing potential Exclusion Criteria: * Performance status \< 90% in patients 70 years old or greater, \< 80% in patients less than age 70 years * HCT-CI/age score \> 5 points (patients with greater than 5 points will be allowed for trial with approval of the principal investigator and the co-principal investigator or his designee; this is an adjustment to account for healthy patients who meet the spirit of this protocol but have histories that result in higher than HCT-CI 5 points; an example is a patient with a solid tumor malignancy in their remote history \[adds 3 points to HCT-CI total\] where the treatment for the malignancy occurred years to decades before and there has been complete recovery of toxicities) * A diagnosis of chronic myelomonocytic leukemia (CMML), unless in morphologic CR * Human immunodeficiency virus (HIV) positive * Active involvement of the central nervous system with malignancy * Inability to obtain informed consent from patient or surrogate * Pregnancy * Patients with life expectancy of =\< 6 months for reasons other than their underlying hematologic/oncologic disorder * Patients who have received alemtuzumab or antithymocyte globulin within 8 weeks of the transplant admission; the absence of these therapies in the medical record will serve as documentation that they were not given * Patients with evidence of another malignancy, exclusive of a skin cancer that requires only local treatment, should not be enrolled on this protocol