If my baby has a dirty diaper on Shabbat and it stains the clothes, can I rinse the clothes under the faucet to prevent staining and to prevent he smell from permeating the home?

It is forbidden to wash clothing on Shabbat. Including in this is pouring water on them or soaking them in water.

If the smell bothers you, try leaving them outside in the yard or porch or hang them out the window.

Sources: Mishna Berura (302: 53, 55), Shmiras Shabbos Kehilchsa (ch. 15: 4)

See Gemara Shabbat (32a) “תנו רבנן: על שלש עבירות נשים מתות יולדות, רבי אלעזר אומר: נשים מתות ילדות. רבי אחא אומר: בעון שמכבסות צואת בניהם בשבת”

All the best!