The term "capital base" has various connotations in the finance world, but, in general, it refers to a base level of funding. For those companies who just went public in an initial public offering, for example, the capital base is the money acquired from the offering, plus any accumulated earnings made through regular sales.
Banks provide a particularly nuanced case. For banks, their capital base is their assets minus their liabilities. Simple enough. They are required to keep a certain amount of capital on hand in order to make loans and service their customers.
But sometimes the value of the collateral backing up a bank's loans goes down, threatening the amount of the capital base. For example, the value of local real estate may have tanked or interest rates may have gone way up. As a result, the amount being paid back on loans is less than the amount the bank needs. The bank will have to raise funds by issuing bonds, reducing expenses or increasing their assets.
"Capital base" also refers to opening an account in a brokerage firm. Your capital base is the money you put into your account in order to purchase securities. Future contributions to the account add to your capital base (as does any profits you make on your investments)