Turning your modem off at night won't cause any problems, though it can be a hassle, especially when you forget to turn it back on and wonder why the internet's not working. (I've done that a few times.)
Now, as long as you are connected to the internet and your computer is on, there is a risk of you being hacked. Thus, some would advise you to turn it off at night, along with your computer. However, this is not necessary, just an extra precaution, as a good firewall, especially one that's hardware-based, should protect you.
Another factor is if you turn your computer off at night. If you do, and it's clean of malware, you can leave the modem on without worry. However, there are some types of malware that can set your computer to automatically turn on at, say, 3:17AM local time, communicate with the sender, and turn off before you wake up. Nice of them not to wake you, isn't it? Therefore, if you're completely clean of malware and have a good security suite, you should be fine, but you can take the extra security precaution if you want.
Hope this helps,
P.S. This can also happen on dial-up, athough it would require you to either have stored the password or the program to have the ability to record your keystrokes and do it for you, which is a little trickier.
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