Dampen a clean rag with some mineral spirits and rub the travertine tile down to remove dirt, dust, oils and other build-up that could affect the finish of the paint. Pour enough mineral spirits on the rag so that it is wet but not dripping.
Rinse the rag thoroughly with clean water, and then wipe the tile again. Allow the tile to air dry.
Apply a layer of stone sealant to the travertine tile, and allow it to dry for up to 30 minutes. This will allow you to apply the color without having it sink into the tile.
Apply a thin, even layer of latex paint to the travertine tile in your preferred color using a paintbrush. Allow the color to dry, and add another coat if the color isn't as intense or solid as you'd like.
Add another coat of stone sealant on top of the colored tile, and allow it to dry.