St Goar is a good place to stay, generally speaking.
The best scenery on the river lies between Bingen and St. Goar. The K-D schedule shows that MOST boats leaving Bingen (which they do at 9:15, 11:15, and 14:30) terminate in St Goar. After that they turn around and head to Bingen. So... cruising to Boppard or Koblenz would mean a LAYOVER in St. Goar and then boarding a second boat that travels from St Goar later on to Boppard. This added cruise adds time and cost to your journey, and IMHO is not necessary. And since your hotel is right there in St Goar, it seems a good idea to end your cruise there. When you get off, you just walk across the street to your hotel. Perfect. Here's a picture of the Hotel Rheinfels, with the K-D boat dock right in front of it.
Your plan was to pick up a car at FRA airport, then you "Wanted to leave car in Bingen and cruise" north from there. Two thumbs up from me to the idea of seeing the Rhine Valley for the first time by cruise boat. But it is indeed a hassle to backtrack to Bingen for the car. And you will pay for parking in Bingen for the time it takes to cruise and get back by train. And driving jet-lagged or drowsy after a red-eye flight is wrong. So I also like the idea of just taking the train from FRA to Bingen... then cruising directly to St Goar with your bags. You can pick up your car for the rest of your trip later, maybe on your 2nd, 3rd or 4th day, depending on whether you need one on those days.
This train thing might sound tricky, but it's not. There's an underground train station right at the airport called the Regionalbahnhof for this trip of about one hour. Every half hour you can catch a commuter train from there (at 8:00, 8:30, 9:00, 9:30, etc.) to the city of MAINZ, where you arrive 25 minutes later. And from Mainz station, there are direct trains to the "Bingen Rhein Stadt" Station (leaving Mainz at 8:32, 9:02, 9:34, 10:02) which also take about 25 minutes.
FRA > Bingen costs €11 each. Since you don't have to return to Bingen by train for the car, or park your car there, that €11 is the only cost.
The K-D ticket kiosk and boat dock is only a 5-minute walk from the Bingen Rhein Stadt station. There are signs at the stations, like this one, that show upcoming train departure times, destinations and track (Gleis) numbers.
Another advantage of using the train to reach Bingen is the 20% discount you get off the cruise ticket. Keep your train ticket and present it at the KD dock for this discount.
KD welcomes passengers with hand luggage on board... no big trunks or similar, though.
I imagine you can plan to catch the 11:15 boat in Bingen, which will put you in St Goar at 13:15, just about the right time for lunch and checking in. Maybe see Rheinfels Castle that afternoon.
St Goar is a good place for transport. Trains go north and south from there, and you can use the ferry crossing to reach St Goarshausen on the opposite riverbank, where trains run north and south on that side as well. Rüdesheim and Braubach (Marksburg Castle) are common destinations. So you should be good without a car on Day 2, maybe on Day 3 as well.
I have previously picked up cars in Koblenz and Wiesbaden (which can both be reached by train from St Goar.) Maybe Mainz will work for that as well. My only interest anymore is train travel, but you should look into one of these places, if you want to pick up a car for the rest.
