In practical terms, it shouldn't. Market cap changes every day (assuming public trading, of course) or even second-by-second, and focusing on investor sentiment toward your company's stock is not the wisest way to make strategic decisions.
That being said, company execs do need to be mindful of unusual swings in their company's share prices because it can sometimes be an indicator of news/information of which they're unaware. At the same time, you can't just disregard your shareholders, especially the big institutional players who may have large voting blocks with which to replace you if they feel you're not responsive to events. They are the ones who make strategic decisions based on your company's share price, right? (grin)
The issue around swings in market cap is more about public perception than reality, so it is important for companies to have a good public relations strategy ready to go that can address questions/concerns in case of some market event. After all, consumers who hear that a company's share price has suddenly fallen by, say, 30% might be more hesitant to do business with that company because there's a (perhaps irrational) fear the company's not doing well and may not be around much longer.
Investors are, by their very nature, emotional rather than rational. Any kind of news can cause a stampede toward or away from a stock for no reason that an investment professional could ever explain. That's why it's impossible to spend any real time focusing on market cap (leave that to your P.R. department to worry about).
IF, as a company executive, you focus on doing the right things to make your company successful then any questions/concerns about market cap will resolve themselves.