Under the tap on the back patio I have a very large terracotta bowl which is a 'back up' water bowl for my dog. She kindly shares this with the doves.
The other back tap has a sort of concrete bowl with a bird on the side...purely to stop the erosion you mentioned. The front tap is permanently attached to the hose, so I don't have anything under that one.
Cheers,
Dee.

A once garden bed now topped with a thick layer of tiny stones. Looks quite good.

Under all my taps is a large green plastic box that collects any water that drips while the hose is running. I use the excess water on my pot plants or bits that I have missed with the hose.
If your tap is dripping get a plumber in to fix it, or replace the washer yourselves. If it is as soggy as you describe you need to do something about preventing the problem rather than dressing it up.

Bromeliads, ferns and Canna Lillies and when the clumps get too thick I pull them up.
Try fighting your way though Canna Lillies to get to the tap at times is no fun.
I may think about pulling those Canna's out now that the Broms are thickening very nicely.
Cheers..MM.

lucky bamboo in a pebble filled vase, it takes the soap and everything. Next to that is giant mondo, cordy's and ajuga reptans

Alas under my tap are all the plants I put there to remind me to repot and repair. I put them there knowing I will see them and remember they need rescuing. Typically I turn the tap off walk away and forget about them until next time.

a new drainage pipe and some concrete would work wonders,if the tap is in a low area.

garden_worm

All our taps have hoses attached but there is a garden with a layer of stones either side of the house. This was advised by the pest man. I have pots standing on the stones to brighten them up.

A pot with mint and lemon balm does well under a tap and keeps them contained as long as you cut any roots that grow out of the drainage holes.

Cheers, Jules

Thanks everyone.

Yes Cattus, I forgot to mention, a collection of plants-in-waiting is also a feature of my tap area :-)

I'm thinking pebbles, stepping stones, mondo grass. I'd love ferns but most I see are labelled frost-hardy or heat-hardy, never both, so I've never experimented. I like the idea of a decorative bowl but I often put a bucket down there when I'm splashing around. Maybe I could make a bowl-like structure, with an overflow drain to share the water with plants further down, out of concrete, custom made to hold the bucket when required. Probably not strictly necessary for drainage but it could look nice. I saw some lovely tiled concrete birdbaths on one of the US forums. It's encouraging to know that utilitarian areas like under the tap don't have to be boring. Suddenly I'm seeing all sorts of possibilities.