"...would you recommend renting a car vs. train/public transport?"
No, not generally speaking, not unless you pick accommodations that would be difficult without a car. An example might be the clifftop-situated Auf Schönburg Castle-Hotel in Oberwesel, or one of the winery-inns tucked away back from the Rhine on one of its tributaries.
Here is the VRM Guest Ticket information that Sam refers to.
You can see the train routes for this part of the Rhine on this map. It also shows the ferry crossings (trains only cross the river in Koblenz.) Note that train users will probably find St. Goar the most convenient base town if they plan to use the east-bank train line that connects Braubach (home of Marksburg Castle, offering the best Rhine castle tour) and Rüdesheim (famous and touristy wine town.) Rheinfels Castle, scenery, central location, waterfront hotels, and cruise logistics also make St. Goar the top choice, IMHO.
Keep your eye on the hotel bookings. Some may not yet be open to reservations. But there will be a lot of demand for rooms that weekend in all the towns. The Rhein in Flammen is in St. Goarshausen, across the river - so St. Goar will surely be popular.
Here's the St. Goar TI office set of listings - not 100% complete but pretty close.
The Guest Ticket for two is inexpensive if you must buy it - but it is also available for free at a few locations in St. Goar, including this apartment / guest house, operated by one of Rick Steves' recommended hosts:
The "booking" engine I often consult says you can't reserve yet; you may want to contact them directly at their homepage above if interested.
Boppard is also a very good choice on the Rhine - and although you don't mention the Mosel specifically, perhaps a preferable base town if you are going to take outings by train to the Mosel towns and castles (that's where Eltz and Reichsburg are located.)
Boppard has more wine spots as well, I'm pretty sure...
List with addresses: https://www.boppard-tourismus.de/files/was-ist-wo-broschu__re_dt_190219winzer.pdf
TI Boppard rooms/apartments
Final thoughts on driving...
Driving from the airport into unfamiliar surroundings after a red-eye flight across several time zones - in a possible state of jet-lag - is a really bad idea.
Driving from place to place on the Rhine will but the vise grips on your wine fun.
Baden-Baden: You don't need a car to get there. Here's a map showing Baden-Baden on the major train line into Switzerland.
Also... if you are not into casinos and spa treatments, go somewhere else! There are many nice places in the region that are great to visit and easy to reach by train. One option: ake the train a little further south - to Offenburg - and change there to the Black Forest Railway (like the Rhine, of of Germany's most scenic routes.)
Gengenbach (near Offenburg on that map) is an old walled town and a great travel base.