Technically yes you can replace the railing with a wall. But it's more complicated than a simple swap, you'll likely have to alter things like HVAC, electrical, support etc.

There are probably some small changes you can make to soundproof the living room or the bedrooms. Maybe try that before jumping into this.

Whatever you do at the rail must be reversible.

Making that into a giant featureless wall will also have an impact in how you perceive the living area. And be a bear to try to do anything with.

This is what happens when form follows fashion instead of function.

There are a lot of things to consider, so many that you really need on the spot advice. Look around for a good architect or builder that's experienced with the shortcomings of open concept McMansions and have them come out, take a look and talk with with you about your options.

If you decide to do nothing, your kids will probably adapt to the level of noise and sleep right through it.

For his first 6 months of his life, my baby brother's crib was right in the living room which had the TV. To him, I guess it was the sound of home! (My dad was in the process of building a bigger house.) And my own first house was close to a railway track - my infant son slept right through the early morning commuter trains, and after awhile I did too.