If someone eats blue cheese (which is manufactured with some variety of the penicillium mold) while they are sick, does this a have any effect on creating antibiotic resistant bacteria?
Well, yes, but don't worry.
New antibiotics that are active against resistant bacteria are required. Bacteria have lived on the Earth for several billion years. During this time, they encountered in nature a wide range of naturally occurring antibiotics. To survive, bacteria developed antibiotic resistance mechanisms. Therefore, it is not surprising that they have become resistant to most of the natural antimicrobial agents that have been developed over the past 50 years. This resistance increasingly limits the effectiveness of current antimicrobial drugs. The problem is not just antibiotic resistance but also multidrug resistance. [From: Microbial drug discovery: 80 years of progress]
That means the Penicillium roqueforti is in a constant fight with bacteria since 'forever' using its antibiotics to gain the upper hand when used in cheese. Those bacteria responsible for any infection coming into contact with those antibiotics and surviving the encounter will be somehow more resistant to those chemicals than before the encounter. They may have very well been resistant to those before.
These survivors may be naturally resistant to the antibiotic ingested, or they may have acquired resistance during therapy. Sometimes the patient was already colonized with resistant organisms before treatment had even begun.
And this is of a very small concern because blue cheese mould are not producing antibiotics or other chemicals of use as a medical agent for pathogenic human infections with bacteria. Those chemicals are either very weak or in very low concentrations.
Penicillium is a very large group of species and not every member of this family produces useful drugs. Resistance to antibiotics is an evolutionary process and evolution favours minimalism in regard to developing new features. Eating these cheeses, if properly prepared and stored, should not increase resistance against prescription drugs we use currently under the term antibiotics.
So, after detailing some very nit-picky technicalities let me clarify my opening statement to: No. These cheeses will not cause problems with resistance to prescription drugs.