Nash social welfare function with negative exponential utilities
I completely agree with @FooBar. Of course the utility function can be negative in some of its range. Conceptually, one could invoke the cardinality vs. the ordinality argument as a response to your question. I would think that your problem is as follows:

$$
\
max\, SWF=u(x_{1})\times u(x_{2})\times...\times u(x_{N})
\
 s.t
\
f(x_{1}....x_{N})\leq g
\
$$

The constraint(s) can be some resource constraint. All that matters here is that the SWF satisfies the normal properties associated with the existence of a maximum on the constraint set (You can invoke the assumptions of the Weierstrass Theorem here). Moreover, you should check Second Order Sufficient Conditions (in this case the negative definiteness of the Bordered Hessian) for a maximum. The value of the welfare function by itself is irrelevant as long as it is a maximum on the constraint set.