If you google understair storage ideas the are tons , here we need to know what you want to store that is really the fist step

Built-in shelves all the way up under the stair is the obvious answer. What's that door on the right?

Make it a bedroom for an adopted nephew.

Thanks RappArchitecture and Patricia Colwel consulting for your quick and helpful response.

The door to the right is a toy closet. I’d like to make the rest if the area under the stairs able to hold more buckets for toys or a small riding toy etc.

a few houzz photos...

Under Stair Storage · More Info

2013 Coastal Living Showhouse · More Info

Pfeifer Farmhouse Wine Cellar · More Info

Family Farmhouse Addition · More Info

Beckwith · More Info

Ideas

I can cut out more sheetrock to mane the opening a but larger but it what will be the nicest way to make a door of some sort to cover this new opening?

*a bit larger

The questions that come to mind are (1) is there anything below this (i.e., a basement)? (2) what is that door on the right?

A certain amount of structure is required to hold up your stairs, but beyond that you can put in any number of doors or drawers. Sometimes people make the risers of their stairs into drawers, but from the side is natural too.

Contemporary Staircase · More Info

Indigo Tiny Home · More Info

Grove Avenue · More Info

Michigan Summer Home · More Info

Under Stair Storage · More Info

Lots of options, but be sure to get some advice on what supports you need to work around before trying anything like this yourself!

You are correct. contractor says those support beams need to stay.

so Now, whats the best way to maximize this space?

I think the making some sliding drawers , with upper shelves , that fit within the defined areas of the support beams will give you so much more storage. Note that a couple of the example images appear to show pull out drawers that have supports between the pullouts.