I think painting is your best bet at this point. I'm sure someone will tell you what type of paint works best on stairs....

you can use c2 paint cabinet trim they will add an additive to be used on floors

Be very careful about buying flooring from big box stores as they sell seconds and rejects. I've been doing a lot of research in the last few months deciding what solid wood floor to go with, and that's what I learned from the pros here and on theflooringpro.com's forum. You might go over there and ask them what the recommend. They were very nice. And go talk to your local flooring store. You want a quality product, especially on stairs where your kids could trip and fall.

Great idea Margo. I just took a peek over at their website and the variety and prices are amazing. The warranties are prerry good too. Got me to thinkin' !

Definitely paint.
You can continue to sand - use 140g sandpaper with a Makita handheld sander.
If that does not work, simply sand any rough edges to smooth them down then paint with a primer (interior wood primer)which is for pre painted wood . Then one or two coats of interior wood paint (for floor) as it's suited for heavy traffic.show this beauty off!! Good luck with your project .

I would also add that running on wood stairs with shocks is a bit of a slip hazard; in which case, a runner sounds good as well (after painting).

If you do decide to paint, go for water based primer and top as it will dry quickly while children are at school ;)

Socks even ;)

Can you put wood on the back of each step? Then you can paint the top and have the bottom painted light like in this picture. Then you aren't always looking through to the weird tile hiding space below.

We recently added stairs to a bonus room (basic wooden stairs similar to the ones you have). We painted them with an outdoor deck paint because we have two boys who will be using the stairs a lot. The deck paint should hold up well.

It looks like stair is going down to finished basement or lower level. With 3 little boys and heavy traffic, first and foremost I would bring this stair to code and install risers.
With heavy traffic paint will start to peel and you will gave bits and pieces all over the place, not to mention it will make a slippery surface unless you add additives to make it slip-proof, they sell special stuff you add to paint to achieve that.
Best bet is to wrap the stairs with durable, stain resistant commercial carpet it will last for many years. They have nice Berber carpet on the market, it is not that expensive and its made specially for high traffic areas.

Good luck.