When using logistic regression, you are fitting the following linear model:

log(p/1-p) = b0 + b1*x1 + b2*x2 + ... + bk*xk + E

As you can see, you are modeling the changes in the log odds. That's usually not very helpful. When analyzing this type of models, it is often more informative to obtain odds ratios, which are the exponentiated coefficients. And there's no odd ratio for the intercept, so it won't matter whether is negative or not. You can safely remove it from the model if not significant.