The price would go down until it finds enough buyers for a given price and there will be a price for which the stock is attractive again. For instance, if a company raises capital by doubling the volume of its shares, at half the price the company will be worth the same so there is no reason why nobody wouldn't buy shares of the stock at half the price. Now if there is for some reason 0 buyers, the stock is illiquid and the stock price goes to 0 but the company can still issue new shares.
The shares are issued on the public market so market participants would buy the shares. Order book and market orders will determine at which price these shares would be bought.
That would be true for private companies but I'm not sure what would prevent public companies to raise capital by issuing new shares. Obviously the stock price would take a hit but new capital would be raised. That's the function that public markets are supposed to fulfill.