Remove any loose caulk,
Use painters tape (blue stuff) to tape off the edges of the stair tread. Be sure to continue over the bullnose front. The tape will give you a nice straight caulk line and protect the nice treads from getting caulk on them. ( I like to have tape covering at least 2 inches.)
Apply new latex paintable caulk. Smooth the caulk with a finger then wipe with a damp Grout sponge. The bigger sponge gives a nicer finished line.
Remove the tape soon after wiping, before the caulk fully dry or you may pull the caulk away.
When the caulk is dried you can touch up the paint. Applying tape again will prevent paint from getting on the treads.