It is possible to watch New York Yankees games live online or on TV without an expensive Cable TV package. Usually these games are on YES, but that may change depending on schedule, type of game or national interest.
The New York Yankees play most of their games on YES, a regional sports network. Regional Sports Networks (RSNs) are not common on streaming services, but when they do exist they are usually only available for an additional fee.
The MLB also has MLB.tv, which live streams all out-of-market games for all teams. Unfortunately, New York Yankees games on here are subject to blackouts if you are within what is considered the team's "home region".
If you find yourself trying to watch a game on the MLB.TV that is blacked out, you will need a VPN to mask your location.