(1) Background: Sports games are one of the best ways of engaging in physical activity for individuals with intellectual disabilities (ID) and Down syndrome (DS). This systematic review of the current literature aims to identify and sum relevant data on motor skills and clarify whether there are positive effects of sports programs in motor skills games in children, adolescents, and youth with ID and DS. (2) Methods: The systematic review of the papers was carried out following the methodological guidelines and by the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) consensus. The following electronic databases were researched: PubMed, MEDLINE, Google Scholar, ScienceDirect, and ERIC from 2001 to 2023. (3) Results: The basketball test battery can be used to improve and monitor basketball training. Basketball players with lower levels of ID achieved better results, especially those with disabilities of the II and III degrees. Futsal has a positive impact on the coordination, and the differences between the test results of the vertical jump with and without an arm swing, that can be seen indirectly as a coordination factor, were statistically significant. (4) Conclusions: Basketball is recommended as an effective and practical rehabilitation program for children, adolescents, and youth with ID and DS. Futsal is an interesting and helpful activity for individuals with ID as well.