Basel train station 30 minute connection

30 minutes is plentiful: 15 minutes would be enough even with a family in tow.
It gives you a bit of buffer should you train from Paris be a bit late.

More than enough, but I hope you don't plan to buy Paris-Basel and Basel-Lauterbrunnen as separate tickets.

Good point just above. SNCF won't sell you a ticket to Lauterbrunnen, but it will to Interlaken Ost. You can then buy a ticket to Lauterbrunnen separately. Check the options!

Yes, 15 minutes is OK, 30 minutes is plenty.

All the platforms at Basel SBB station are parallel. There are escalators up from each platform to a big bridge structure, and down again to all platforms. Try to find out the correct platform number before you get off your train from Paris, for example from the SBB App. Or look at one of the YELLOW video screens once you get up to the overpass.
There are shops on the overpass, you will have time to grab a coffee or eats.

Station Plan: https://www.sbb.ch/content/dam/infrastruktur/trafimage/bahnhofplaene/plan-basel-sbb-a4.pdf

There is no need to buy two separate sets of tickets. The SBB website will sell you tickets from Paris Gare de Lyon to Lauterbrunnen. From Paris to Basel these will be train-specific tickets, with train ID and seat numbers on them. From Basel to Lauterbrunnen you can travel on any train on that date. So there is no train to miss!

You will also need to change trains at Interlaken Ost.

More than enough, but I hope you don't plan to buy Paris-Basel and Basel-Lauterbrunnen as separate tickets
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I was following the information from Seat61.com. I was planning to buy Paris-Basel on SNCF and Basel-Lauterbrunnen on SBB using the Swiss half fare card & the family card. The kids would travel for free from Basel to Lauterbrunnen with the free family card.

I would greatly appreciate any information on how to get from Paris to Lauterbrunnen by train.

It was my understanding that purchasing tickets on the Swiss train website was a more expensive option and that my plan above would be cheaper.

Right. If you are planning to use a half fare card for Switzerland, then are you really getting significant savings on the "supersaver" train-specific tickets from Basel to Lauterbrunnen? You might save 10 CHF per paying adult, if that. I would get regular half-fare tickets, which are valid on any train of the day. That way, no stress and no worries!

If I purchase Paris-Lauterbrunnen on SBB it's $460 total (that's with the 1/2 fare card and kids traveling for free). If I split the tickets and buy Paris-Basel (SNCF) & Basel-Lauterbrunnen (SBB) the total is $204.

This is for the exact same ticket-same dates, times, trains.

The connection in Interlaken Ost may be a little tighter. We had to run for our connection, down, across, back up etc.. Of course, there is always another train within 30 minutes. With kids in tow, it might be easier just to wait for the next train.