Selective Depletion of CD45RA+ T Cells From Allogeneic Peripheral Blood Stem Cell Grafts in Preventing GVHD in Children

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 * Acute myeloid leukemia in relapse or primary refractory disease with a circulating blast count of no more than 10,000/mm\^3 * Refractory anemia with excess blasts (RAEB-1 and RAEB-2) * Chronic myelogenous leukemia with a history of accelerated phase or blast crisis * Other acute leukemia (including but not limited to 'biphenotypic', 'undifferentiated' or 'ambiguous lineage' acute leukemia) * Patient with a human leukocyte antigen (HLA)-identical (HLA-A, B, C, and ribonucleic acid \[RNA\] binding motif protein 45 \[DRB1\] molecularly matched) unrelated donor or related donor capable of donating PBSC * DONOR: HLA-matched unrelated donors (HLA-A, B, C, and DRB1 matched based on high-resolution typing) capable and willing to donate PBSC * DONOR: HLA-matched related donors \>= 18 years and capable and willing to donate PBSC 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 * Patients who weigh \>= 70 kg must be discussed with the principal investigator prior to enrolling on the protocol * Patients who are human immunodeficiency virus positive (HIV+) * Patients with uncontrolled infections for whom myeloablative hematopoietic stem cell transplant (HCT) is considered contraindicated by the consulting infectious disease physician (upper respiratory tract viral infection does not constitute an uncontrolled infection in this context) * Creatinine \> 1.5 mg/dl * Cardiac ejection fraction \< 45% * Patients who can perform pulmonary function tests will be excluded if they have a diffusing capacity of the lung for carbon monoxide (DLCO) (corrected for hemoglobin) of \< 60% predicted; 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 * Patients who have liver function test (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 protocol 2660 is contraindicated for that patient the patient will be 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 protocol * 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 * DONOR: 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 * DONOR: Donors who fail eligibility requirements for donation of cells or tissue per section 21 Code of Federal Regulations (CFR) 1271 for donation of a HCT/product (P) will be excluded unless use of the cells complies with 21 CFR 1271.65(b)(iii) (urgent medical need) or with 21 CFR 1271.65(b)(i) (allogeneic use in a first-degree or second-degree relative) * DONOR: Unrelated donors donating outside of the United States of America (USA) or Germany