Not knowing the moisture levels in your basement, I would recommend tile floors. The grout lets the moisture breath through and up into the house. If you basement is totally dry, these would still work. You can put relatively inexpensive rugs over the tile for comfort and change them out as needed over the years. With the variety of tiles out there, the sky is the limit, you can even get ones that look like wood, I'd stick with something light like this example from Home Depot: Montagna Dapple Gray 6 in. x 24 in. Porcelain Floor and Wall Tile

What is the floor you have inside? Tile? Don't laugh, I'm in S Fl where we have lots of tile. Wood? Wood-look tile? Something else? I have always thought that it makes a wonderful extension of your current living space by having the same flooring extend from your living area to the covered patio area. Of course, this only works with certain types of material. Something like the concept below where the folding doors (or sliders) hide completely and the floor extends the living area to the outdoors. ETA: If you can't use the same material, either use a totally different material or use something that has the same color so it blends. Without knowing what you have on the floors in your current home it is hard to say. Here is one where a different material was used extensively for their outdoor areas:

The most budget friendly would be a sharp, spade cut, traditional slit cut shallow trench maintained with edging shears.

We agree with Patricia. Matching the wood floor that is already there would make the room feel bigger and more congruous. A doormat outside and a remove your shoes, please, Policy is always great with wood floors or any floors for that matter. Although not always possible. Lumber liquidators is a great place to go to for wood flooring of all kinds. As asked earlier; what is your plan for this room?