(1) your operating system (XP? Vista? Windows 7?)
(2) exactly what modem you got
Here are some generic ideas:
(1) try reinstalling the modem driver. a CD containing it probably came with the modem
(2) try moving the modem to a different slot, if there's an empty one. maybe the slot has gone gimpy
(3) try replacing the BIOS battery. if your BIOS battery has died, some system settings may not survive when you turn off the machine, so the machine may fail to remember what hardware you have. advice here would depend on what machine you have, since different machines have different ways of accessing the BIOS and different BIOS options. the fix here would be to replace the battery. to try to decide whether this might be the problem, you need to access your BIOS (probably by tapping DEL or ESC or F2 when the machine boots) and then looking to see whether there's a setting for the modem. Caution: If you've never ever looked at the BIOS, this might not be the ideal time to start
(4) try booting a linux distro on your machine. mint or ubuntu probably will work; knoppix generally works if nothing else will. to do this, you need another machine with internet access, and (say) a 2gb thumbdrive: download
Linux Live USB, install it, and follow the instructions to create a bootable usb. then you need to boot your machine from the thumbdrive; this will require entering the BIOS and changing the boot order so your machine tries first to boot from a usb stick, if one is present. you should then be able to run linux on your machine WITHOUT INSTALLING IT. with any luck, the linux distro will be able to run your modem (but if it can't, you'd have a real headache trying to figure out how to make it happen). if your modem works sometimes, but not always, you might suspect a gimpy slot or maybe the new modem is a lemon
<on edit:> you might research your modem before trying this: say:
http://www.linmodems.org/(5) if you're sure the BIOS is set right and your BIOS battery is good and the modem is good and it's in a happy slot, maybe you should worry about your operating system being corrupted, either because oldjunk files are everywhere or because you've got some virus or malware. don't take advice from me about that: ask the windows users here what they would do. the bleak end of the options is comnpletely reinstalling the operating system AFTER BACKING UP ALL YOUR FILES.
(5) if you've gotten to the end of the line, and you've reinstalled the operating system, and everything worked for a while maybe before crapping out, maybe the hard drive is dying: some system settings may be written to gimpy portions of the harddrive and may not survive there. the fix here would be to replace the harddrive and then reinstall the operating system. there are diskcheck routines you can use to check the disk first
(6) finally -- if you're having somebody else do the work for you, beware of "in for a penny, in for a pound"