Open Source Licensing – Is Any JavaScript Code Used Only for Websites Open Source?

My specific question is "how do you select which one you want to use." In my case, I want to only use the BSD license and I want to make it explicitly clear that I do not opt into the GPL v2 license in any way.

You just do. Unless you don't change the source-code there, you don't even need to touch the plate. It can be stayed as is, yes, even touching it for no reason might look more suspicious then just leaving it as-is.

The licenses are offered to you btw. Into which one you opt, you can say anytime, you can choose anytime which one. The one day you might like BSD and the other day you might even use GPL. Both is possible. You can even choose both, even if they both would be incompatible to each other. Only based on the fact the original author(s) did dual-license it.

How do you do that and have it hold up to legal challenge?

No-one can predict the future, but keep references of the websites of today, so that you can document that you've taken care and you can back-up your statement that the dual-license is disjunctive (OR) and not conjunctive (AND).

As long as it's clear that you were allowed to use the software under the terms of the BSD license (and you fulfilled the requirements) I see no reason why it shouldn't hold up a legal challenge, I can not even imagine of which kind such a legal challenge would be?

This is not a clear-cut issue. Consider two extreme ends of the spectrum:

1. Your proprietary client software is an HTTP client and it renders HTML responses. It can work with any HTTP server. The HTTP server that you use for your service happens to use GPL components.

2. You have a program that uses GPL-licensed components. You pick an arbitrary point in that program's operation and break the program into two programs. The two programs communicate over a totally superfluous network hop. You put all the GPL-licensed components in the first program and license under the GPL, and you license the other program under a GPL-incompatible license.

The first case is clearly okay. The second case is clearly not okay. You haven't given much information about your particular case, and even if you did, only a court ruling could decide definitively whether you're in the right.

The GPL FAQ has this to say on interoperable, separately-licensed programs:

However, in many cases you can distribute the GPL-covered software alongside your proprietary system. To do this validly, you must make sure that the free and non-free programs communicate at arms length, that they are not combined in a way that would make them effectively a single program.

The difference between this and “incorporating” the GPL-covered software is partly a matter of substance and partly form. The substantive part is this: if the two programs are combined so that they become effectively two parts of one program, then you can't treat them as two separate programs. So the GPL has to cover the whole thing.

You must decide whether you think your client are server meet the standard of "two parts of the same program" (and therefore must each be licensed under the GPL) or not. The GPL FAQ gives some further explanation on this topic on another question:

Where's the line between two separate programs, and one program with two parts? This is a legal question, which ultimately judges will decide. We believe that a proper criterion depends both on the mechanism of communication (exec, pipes, rpc, function calls within a shared address space, etc.) and the semantics of the communication (what kinds of information are interchanged).

...

By contrast, pipes, sockets and command-line arguments are communication mechanisms normally used between two separate programs. So when they are used for communication, the modules normally are separate programs. But if the semantics of the communication are intimate enough, exchanging complex internal data structures, that too could be a basis to consider the two parts as combined into a larger program.

So, network communication certainly passes the "mechanism of communication" test but it is unclear where your client/server pair falls on the "semantics of communication" test.