Ooops, double picture. Last one Should have been this:
Facing Front of House.

That support beam is used to provide the support without a wall having to be built where it is. In other words, putting a wall in place of the support beam---built entirely of 2x4's---would provide as much or more support. But a wall does not allow the access to open space posts and headers do. The header transfers the weigh above it to either side the opening under it.

If you wish to provide support for the dining room table, that wall for the laundry room is enough---except where the opening is---there needs to be a header there. A double 2x8 header would be sufficient, IMHO.

Thanks handymac. I plan on making the laundry wall smaller. Instead of a 72" door opening, it will be about 42". So the wall won't extend out like you see.

I'll have to add a header on the other side of it. Do I use support brackets to secure it to the floor joists?

Just nail the top plate to the joists---and use pressure treated 2x4's on the concrete floor.

I have to slightly disagree with handymac. The wall that you are putting up in the basement has a single 2x4 top plate and the studs are not aligned with the floor joists from the floor above. This is going to add very little additional support, though it will be better than nothing. By the picture, it appears that your joists are 2x10's on two foot centers. On a 13' span, this meets code with almost any wood species.

If you are not too far along with this project, drop that new wall, use a treated lumber sole plate, install a double 2x8 header on your openings with two studs on each end of the header and a install a double top plate. You'll be fine.

I did take that wall down because I thought it was to big for the basement. So what I'll do is "FIRST" add the header all the way across. I got 18 feet of it.
Then install 4x4's on the side of the header down to the floor. Under each joist I will add some 2x4's on center.