I don't know which online casino we are talking about, but I would venture to say that online casinos, in general, are probably not the most trustworthy of businesses. Caution is certainly in order. That having been said, this isn't an e-mail from a stranger that contacted you out of the blue; you obviously trust them enough to have deposited some money with them, and it seems that they now owe you money.
Let's assume for the moment that they are legitimate, and that they sincerely want to pay out your winnings. If they are to pay you via a wire transfer, they would need your account number and routing number. (This information is on every check that you write.) In addition, if this is an international transfer, they would also need your bank's SWIFT number, or possibly an IBAN code.
It does seem odd that they would pay you a partial payment with a check, but the rest has to be done via a wire transfer. You could request that they send the remainder as a check, but I would imagine that if they refuse to send you a check, there is nothing you can do about it.
If you decide to go ahead with the wire transfer, you could open up a new savings account with your bank first. Then you could provide the account number for this new account, and if they are intending to clean out your account, there will be nothing in it. (For extra protection, when you set up the account, you could ask the bank if they can set up a savings account that will accept incoming wire deposits, but no outgoing electronic withdrawals.)
Either way, when you deposit the check you have and you receive this wire transfer, don't spend this money for a while. Just let it sit in your account (you could transfer it to your main account, if you like), and wait a few weeks. That way, if there is a problem with these payments and your bank insists on the money back, you will not be in trouble. If they send you more than they owe you and ask for some of it back, it will be a clear indication of a scam. Don't send them any money back.
After a few weeks, you should be in the clear. Good luck.
By the way, online gambling is a terrible idea. The fact that you don't trust the casino to pay out should tell you a lot about this industry. After you receive these winnings (or even if you don't), the best advice I can give you is to stop gambling.