A proper model selection procedure should be performed in this case. For example, you can use LR test to compare nested models (with and without a specific covariate). You can decide to keep your clinically important covariates even if the model selection procedure doesn't suggest it.
Model selection based on univariate analysis is not recommended. There are a lot of materials on the web for this topic.
A lot depends on your specific research question. If you've set out to evaluate whether a given variable helps improve a model for a categorical outcome variable, then include that variable in the model.
If you're somehow trying to identify an "optimal" subset of IVs (independent variables) for a logistic regression model, then, as Eliana suggests, the approach can matter. Methods such as all possible subsets, lasso (or other penalized methods), and others are available (here's a brief discussion of some of these: Article Comparison of subset selection methods in linear regression ...
The so-called "step" methods (stepwise, forward entry, backward elimination) are notoriously challenged for a number of technical and practical reasons.
Regarding the problems associated with methods like bivariate pre-screening & "stepwise" selection, see the Stata FAQ on stepwise methods and Mike Babyak's nice (2004) article on overfitting.