Can the prescription for anxiety cause any problem with my memory?

Possibly. The short acting benzodiazepine anti-anxiety medications are the most likely to cause memory difficulties. My personal view of this is that they act by means of what is called state dependent learning. If someone has an alcoholic blackout, they will not remember what they did the night before. However, if they are given enough alcohol to bring their blood level back up to where it was the night before they would be able to tell you what had occurred. I believe that this is a similar mechanism occurring in the short acting benzodiazepines. The blood levels peak and trough too many times throughout the day and what is recalled at one point during the day maybe forgotten as the blood level changes to another point in the day only to be recalled when the blood level goes back to the original level it was at when the memory was consolidated in the first place.
Yes. Many drugs for anxiety reportedly "fog" your thinking, even though you may not be aware of it. If that's a concern, ask trusted friends/family members about your behavior. Express your concerns to your doc; there are treatment choices that are less likely to cause such problems.