Donor Umbilical Cord Blood Transplant With or Without Ex-vivo Expanded Cord Blood Progenitor Cells in Treating Patients With Acute Myeloid Leukemia, Acute Lymphoblastic Leukemia, Chronic Myelogenous Leukemia, or Myelodysplastic Syndromes

Inclusion Criteria:

* Age criteria:

  * High dose TBI regimen: 6 months to =\< 45 years
  * Middle intensity TBI regimen: 6 months to =\< 65 years
  * Conditioning regimen selection should be based on the underlying disease, presence of minimal residual disease (MRD), age, co-morbidities, attending physician, and site preference; conditioning regimen will not require stratification of the randomization due to heterogeneity in the cohort of eligible patients.
* Acute myeloid leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia

  * All patients must have acute myeloid leukemia (AML) that is considered best treated by stem cell transplant by the referring physician and the attending transplant physician
  * All patients must be in complete remission (CR) as defined by \< 5% blasts by morphology/flow cytometry in a representative bone marrow sample with cellularity \>= 15% for age
  * Patients in which adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures; these patients must be discussed with the principal investigator prior to enrollment
* Acute lymphoblastic leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia

  * High risk first complete remission (CR1) (for example, but not limited to: t(9;22), t(1;19), t(4;11) or other mixed-lineage leukemia \[MLL\] rearrangements, hypodiploid); or high risk (HR) as defined by referring institution treatment protocol greater than 1 cycle to obtain CR; second complete remission (CR2) or greater
  * All patients must be in CR as defined by \< 5% blasts by morphology/flow cytometry in a representative bone marrow sample with cellularity \>= 15% for age
  * Patients in which adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures; these patients must be discussed with the principal investigator prior to enrollment
* Chronic myelogenous leukemia excluding refractory blast crisis; to be eligible in first chronic phase (CP1) patient must have failed or be intolerant to tyrosine kinase inhibitor therapy
* Myelodysplasia (MDS) International Prognostic Scoring System (IPSS) intermediate (Int)-2 or high risk (i.e., refractory anemia with excess blasts \[RAEB\], refractory anemia with excess blasts in transformation \[RAEBt\]) or refractory anemia with severe pancytopenia or high risk cytogenetics; blasts must be \< 10% by a representative bone marrow aspirate morphology
* Karnofsky (\>= 16 years old) \>= 70 or Eastern Cooperative Oncology Group (ECOG) 0-1
* Lansky (\< 16 years old) \>= 60
* Adults: calculated creatinine clearance must be \> 60 mL and serum creatinine =\< 2 mg/dL
* Children (\< 18 years old): calculated creatinine clearance must be \> 60 mL/min
* Total serum bilirubin must be \< 3 mg/dL unless the elevation is thought to be due to Gilbert's disease or hemolysis
* Transaminases must be \< 3 x the upper limit of normal per reference values of referring institution
* Diffusing capacity of the lung for carbon monoxide (DLCO) corrected \> 60% normal
* For pediatric patients unable to perform pulmonary function tests, oxygen (O2) saturation \> 92% on room air
* May not be on supplemental oxygen
* Left ventricular ejection fraction \> 45% OR
* Shortening fraction \> 26%
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* 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
* History of human immunodeficiency virus (HIV) infection
* Pregnant or breastfeeding
* Prior myeloablative transplant containing full dose TBI (greater than 8 Gy)
* Central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy and/or cranial radiation prior to initiation of conditioning; diagnostic lumbar puncture is to be performed per protocol
* Patients \>= 45 years: comorbidity score of 5 or higher