You don't need DeutscheBahn for this one. Basel has 3 railway stations, 1 Swiss, 1 German and 1 French
For the stretch to Paris there are no reservations needed:
NL:
Hoek van Holland, bike to Rotterdam (there's no trainline anymore, only a metro)
Rotterdam-Roosendaal with a Dutch ticket (online via
www.ns.nl)
Roosendaal-Lille with a Belgian ticket (online via
www.nmbs.be)
Lille-Paris via local trains (online ticket via the SNCF or the ticket machine)
alternatively Roosendaal-Mons, cycle 20km and then the Maubeuge-Paris train
Then Paris-Basel via the SNCF. This you can also do with local trains (TER) taking about 6 hours.
The advantage of using this connection is that there's not a single reservation involved and you don't have to disassemble your bike. It would normally need a night somewhere though.