Trains with children

You do not book trains in Switzerland, you cannot (usually) reserve seats, you just buy tickets from A to B, and then get on any of the frequent trains to your destination.
I am not sure, are you saying you are staying in Interlaken and then travelling on to Mürren on another day, or continuing to Mürren the same day? If the later, it would have been simpler and probably cheaper to buy a ticket all the way to Mürren. Or just buy a ticket from Basel to Mürren at Basel station.
You can buy tickets on the day, from the ticket machines or the manned ticket desk.
The same applies to Mürren to Luzern, and Luzern to Zürich, buy the tickets on the day. Note, I am using the local spellings, which is what you will see on signs when you get here.

The rule is "6 to 16 half price*, so the 4 year-old travels free (no ticket needed), and the 6 year-old pays exactly 50%, unless you have a family pass. The full rules are here: http://www.sbb.ch/en/travelcards-and-tickets/tickets-for-switzerland/children.html

If you want to find out times and prices for Swiss trains, you can do this from the same site (Swiss Federal Railways, SBB): www.sbb.ch
For example, Basel to Mürren, one train per hour (this is the minimum on any Swiss rail line), takes 3 hours, and costs CHF 78.60 full fare, or CHF 39.90 for a child or someone with a half fare card.

We are traveling to Murren the same day.

Thank you! I have already made a lot of costly mistakes - I wanted to take advantage of what I thought would be cheaper tickets by booking as far ahead as possible, but the more I get into it, the more mistakes I see I am making! Thanks again!

One more question - we are traveling on a German ICE train from Basel to Interlaken - do the rules regarding child fares still apply? Thank you, again!

FamilyVacation,
It doesn't matter what it says on the side of the train, SBB, DB ICE, SNCF TGV. Once it crosses onto Swiss tracks Swiss rules apply. If I buy an SBB ticket (with my Swiss half fare card) from Basel to Interlaken I can take any train I like, the colour it is painted is irrelevant.
Anyway, the train crew (driver, ticket inspectors etc.) changes at Basel, and these will be wearing SBB uniforms.

I forgot to say explicitly, there is no discount for early purchase of Swiss rail tickets. It is a fixed price. This is unlike French and German railways (SNCF and DB), where express trains have a discount for early booking. Each railway company has its own totally different rules, which is why people get confused.

Thank you again! I think I also paid more because I booked thru Rail Europe, which has the lower prices earlier on, now I see Voyages is less. Thank you for all of your help!

Never use Rail Europe! Expensive ticket resellers and not an actual train company!

:( Found out too late, ugh!