In theory the price a company's shares trade at in secondary markets is not relevant for day-to-day operations.
Obviously if the intention is to participate in the market directly either by issuing new shares or buying back shares, then the price is relevant.
However the share price is significant in another way particularly for a company which needs to borrow; and borrowing (and lending) is fundamental in banking.
According to fundamental analysis, the market capitalization (price per share multiplied by the number of outstanding shares) should reflect the company's equity (excess of assets over liabilities) and the value of expected future profits.
A low share price implies that the market believes that the company's balance sheet is poor and/or that it does not expect the company to make future profits. (Note the markets have a great record of being ahead of the game in this regard as we can see from the Irish stock prices; share prices were falling long before the reasons became apparent.)
This affects the cost of borrowing in the exact same way as it applies to individuals. If the value of your assets/property/etc. is far more than that of your debts, then banks are far more likely to lend to you and at a lower rate. They will also be favourably inclined even 'though you may not have much in the line of assets as long as you have a secure high paying job (the equivalent of expected future profits).
The converse also holds; if they suspect you have more debts than assets and that you won't be able to earn any money going into the future, then lenders will be understandably reluctant to lend to you.
This is what happened to some of the Irish banks. Their share prices were heading to zero (suggesting that their assets were insufficient to cover their liabilities and that they were not expected to make profits) so it's perfectly reasonable for other financial institutions to fear extending credit to them.
It's fair to say that if a bank can't borrow, then it won't last long as a business.
So in a long roundabout way, I guess my answer to your question is, yes.
