I have basement envy! It seems like a nice, large space! I like your idea of painting wood paneling a darker color than top half. I might choose something a little darker than beige to cozy it up, especially if it's for kids to hang out in. Perhaps paint the concrete floor and use some area rugs? They make fabulous indoor outdoor rugs in many patterns that are quite cheap. I thought about inserting DIY fabric bulletin boards inside the large X's to hang photos, kids' artwork etc. But perhaps they are too large for this? It's a great space for your children!

Forgot to ask, how old are your children?

First: remove all decorative faux ceilings beams if possible.

Second: remove bar

Third: remove decorative X on wall.

fourth: paint walls a modern light grey and all wainscoting white. With a white ceiling.

Fifth: if the fireplace is not going to be used I'd paint the brick a shade or two darker than the walls. Otherwise I'd bite the bullet and resurface the fireplace with a more modern finish.
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Sixth:install a floating hardwood floor: with darker wood stain (grey or black tones).

Seventh: float two couches ( same size) in center of the room. You could use the pole as the corner of an L

Eighth: ground the space with a fun colored run under couch area.

For furniture, I'd go to Craigslist or Kijiji. Nothing too precious down there as it may take a beating. They have lots of kitchen table sets reasonably priced.

@old dog new tricks - My kids are 9, 12 & almost 15.

FYI The fireplace works so it may be used. My husband is very against painting it though. Not sure if the white and grey tones would be complementary to the brown/yellowy tones in the bricks?

Maybe wall color could be a much lighter tone/s of the brown brick. Yellow for accessories? I googled brown and yellow rooms, and many different combinations of the colors came up. Glad to know your children's ages, helps with knowing what tone the room could have. As for furniture, it can take a while, but I have found some fabulous, high quality pieces of case goods and upholstery on Craig's List. Also, do you have any local antique/consignment shops that you could look in to find some nice sofas/loveseats? Ikea also carries some nice affordable pieces.

Just to show you the color combo.

Will kids take food downstairs? If not, try a light color budget grade carpet, plan to replace it in 7-10 years. If there will be food, try a light vinyl tile and consult flooring store for what can be installed in basements.

I suggest you keep the bar if it is plumbed and has a sink already. If not, take it away.

Remove big X's. Then prime the walls with white primer. It will be much easier to pick a color scheme once the yellow and brown are gone.

Slipcovered furniture works well for kids. Try Ikea, Pottery Barn and so on. Major department stores have good sales in summer. Or redecorate upstairs and move the old furniture to the basement in the time honored tradition.

The St. Davids Sq. idea above is an excellent one for neutralizing the space- use the lightest color in the bricks as a guide. Personally, I'd leave the wood as is, paint the ceiling charcoal and stucco walls the same color as the grout in the bricks. Keep the bar as a snack station and set up the small tables and chairs tavern style in that area. Stain the concrete floor; add task lighting, area rugs. Sectionals are great; stackable chairs for the inevitable extra guests.

Could you place a floor plan with room dimensions. I suppose you will have areas for specific uses in the room. I love Candice Olson basements. In her early season she redid a basement and painted everything white with a sprayer and just addes a lot of white paint. It made everything disappearand thenshe fillled the room with pale greys and beiges and browns and whites.....I am sure you can find it online under Candice Basements. It will give you an idea of how camoflauging paint can be. Start there and then you can choose a color scheme as everything goes with white.

Yellow and grey are nice compliments to each other. You could throw in yellow pillows or art work that pulls in together to make it feel intentional.