Soundproofing a Door with a Shower Pump Behind It
I wouldn't completely rule out sounds getting through the floor, and the walls via the air. 

But the door is certainly significant,and you're in the right track with sealing it. Is it solid wood or hollow? If the latter, filling it with foam filler might help, though the cardboard honeycomb in such doors may make it difficult or impossible. 

There are self adhesive heavy foam sheets for reducing noise in vehicles. They aren't pretty but on the inside of an airing cupboard door that shouldn't matter. Some of the stuff sold in big sheets is lighter and less effective so be careful. 

An acoustic hood would be a good idea. It would have to have some ventilation, and should be made of thick plywood or MDF lined with noise absorbing material. Normal foam would help but only a bit.