For sure close it in. I would lay a garden hose around the deck to determine some curvy lines (please don't make straight lined beds), Don't fill it all in, do several groups with separation between them. I picture some ornamental grasses in various heights, with some splashes of colored perennials, depending on the sun exposure, which appears to be full sun, look for drought tolerant, which usually means they will thrive under neglect.However they all need some water and care. Most bushes, once established do well with very little effort. Maybe throw in some big boulders to take up space and they need no care at all. . . Who knows, after you put all the effort into it, you may surprise yourself and not have such a green thumb. :)

I agree about the insulation. It's more important for a GH in cold country than plant beds next to the foundation. I had the same dilemma with the dense, rigid foam insulation that extends out 2 ft. from my GH base. It does reduce heat loss and that was very important, but it's also impermeable, so as much as it would have been nice to plant that space, it really wasn't practical. As it's an intrigral part of the garden, I wanted my GH site to look nice and satisfied that desire by laying timbers 2 ft. out from and around the GH at the outer edge of the foam, then filled in the space with large decorative river stone, except for a deck in front of the door, built over the same insulation, which gives the GH a nice, finished look and stones are heavy enough for me to use a vac to keep it tidy and not pick any of them up. I use the rock filled space around the remainder of the perimiter to hold large, decorative pots of easy to grow perennials, some agumented with colorful annuals and some, for veggies. (A 20 gal. pot will grow a brodacious tomato) So I get good planting depth with excellent drainage, easy maintenance, and an attractive "finish" to my GH. BTW: After the timbers were laid and effectively locked in the foam panels laying on the surface, before we added the stones, we we went all around the GH base and used Great Stuff to fill in every tiny gap, so the foam, that is under the foundation as well as down 2 ft. and then flanges out the same, is for all purposes; seamless. Every little bit helps.

it is open on 4 sides you will still have moisture/rain and it will still get wet. If anything the roof structure will make it worse as it will not get the sun to help dry it out.

Photos are always helpful. Add them in the comments and wait for the grayed out appearance to gain full color before hitting submit. We know almost nothing about this area such as location in the world, location in your yard, goals for the space other than you want something different and it is wet. Why raised planters instead of plants that like moisture? What is the water source? Looking just for curb appeal or some other goal.