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It looks like there are at least three bike parking locations connected to the railways or stations.
The site is a bit confusing for me, having to translate it from German, but it seems that one (or more) of them are free and at least 2 charge 1 Swiss Frank per day, 12 frank for a monthly pass.
That site has e-mail addresses and telephone numbers as well as website links, so you can contact them or go to their site to find more information.
If the link does not work for whatever reason, basel bike parking
got me several other links, mostly to the same parking places.
I found that link in this discussion on parking a bike on a Switzerland forum.