There is NO direct train (it means you have to change)

There is a whopping high surcharge to use the FRench superfast TGV over the border till IN the swiss-and this canNOT be REServed online either.

Now you can choose-use for less a FRench domestic TGV till last stop before border (mulhouse), change onward to UNreserved local trains-Basel-Luzern, OR place you trust in what they can maybe arrange at counter.

Unfortunately it seems that each&every newbee plans like headless chickens and also does not seem to even realise about all the common probs have been treated and answered here a zillion times before. There are also pages about how to do REServations-and the common traps that come with it

Here is guide on how to make reservations.

https://community.eurail.com/train-connections-reservations-47/how-to-get-reservations-105

In Switzerland no reservation are needed on your route.

As it is the last day of a long weekend tomorrow I would guess that all Eurail passholder seats are sold out on the high speed trains from Paris so you will probably have to use the slower regional trains to get out of France.