Some sites say cut off the flower stem - on two of mine I didn't. They get a drink once a week and a spray of a feed from Poundland. They've both grown a baby orchid!
One of the babies has just started to grow a route - only taken it about 2 years.
Both babies are still attached to the parent plant.
I've also had an orchid grow a new flower stalk and a new crop of flowers.
Virtually ignore them really. Water VERY sparingly when you remember! Cut off the dead stalk, but if it's not dead, just leave it and it might sprout again. I've never had a baby one grow, but all of mine come back into flower at odd times, in no particular order and for no particular reason! They don't like direct sun though, particularly now it's so hot.
I only cut back the stem that actually dies off down to the next 'node' and it usually shoots new growth from the nodes lower down. My 'keikis' (baby orchids) that sometimes grow on a plant I leave them there until they grow roots about 2/3 inches long then break them off and replant young ones.My baby growths usually stay on mother plant for about 2/3 months before being ready to separate. Good Luck.