In theory, borrowing dollars directly and borrowing them via a currency swap should cost the same, as covered-interest-arbitrage would otherwise be profitable. In practice bank balance sheet limits and regulation stop that arbitrage, so a persistent gap exists. A negative dollar basis means paying a premium to obtain dollars through the swap market.
The basis widens in stress, and blows out at quarter and year end when banks shrink balance sheets. Central banks opened central-bank-swap-lines in 2008 and again in 2020 largely to cap it.
Example: three-month EUR/USD basis at minus 40 basis points means a European bank borrowing dollars via swaps pays 0.40% a year more than the dollar money market rate, roughly $100,000 a quarter on $100,000,000.
Related: currency-swap, covered-interest-arbitrage