All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad  is His slave and Messenger.
  is His slave and Messenger.
Music that involves the use of musical instruments is forbidden according to most scholars. Therefore, it is not permissible to include it in the games you produce. There are sound effects that can serve as a sufficient alternative to using prohibited musical sounds.
As for the money you earn from your work in programming permissible games, it is Halal (lawful). The inclusion of background music does not affect this, since it is not an intended part of the contract for selling the game to its users, nor is it an essential component of the game itself. It is considered secondary, and it is pardoned in secondary matters what is not pardoned in primary matters, as established in the rules of Fiqh (jurisprudence).
Moreover, you are not engaged in adding the music, nor do you approve of its presence, not to mention that it (music) has become widespread and difficult to avoid.
Allah knows best.