Entering a military government facility using a name of a senior officer?

Not clear on what you're asking, but a couple of things would apply:
• If the entry was fraudulent and they are trespassing, then it's a possible criminal code violation* for that and the false statement* (plus others) - refer to law enforcement.
• Were the access procedures followed correctly on the facility side? If not, depending on the level of negligence it might be an issue of reinforced training to possible punishment.

If your question is "We found someone trespassing, what procedures do I follow?", that's going to largely depend on installation/leadership policy. Non-law enforcement can detain an individual until properly turned over to law enforcement, but local installation policy may dictate limitations. Consult with the installation JAG.
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* 18 U.S.C. § 1382 - https://www.law.cornell.edu/uscode/text/18/1382
* 18 U.S.C. § 1001 - https://www.law.cornell.edu/uscode/text/18/1001

I would think this would depend on the facilities entered, the person that did the name dropping, and what procedures (if any) were violated.

This is really vague...secure facilities usually require a badge or a VAR submitted by that individual's security manager to enter. Or they need to be escorted by a cleared individual.