When I switched over to broadband, I went out and bought a USB ADSL modem but (despite following the instructions on my ISP's website) I couldn't get it to connect. When I phoned the helpline, the technician was amazed that anyone was still trying to connect via a modem. He tried loads of things to help me but we still didn't get anywhere. He then said "Look, this is going to be a hell of a lot easier if you were to buy a router". Somewhat reluctantly I did so but the investment turned out to be well worth the money because I've now got a really reliable connection, with a typical speed of around 6.3Mbps. (Because I've only got one computer, in a fixed location, I've disabled the wireless facility of the router and connected it to my PC via an ethernet cable).
The great thing about a router is that it remains connected to the internet at all times. Since the PC itself doesn't connect directly to the internet there's no 'logging on' and the chances of a dropped connection seem to be considerably reduced.
OK, I recognise that it involves some expense, but perhaps using a router might be a better option for you as well?
Chris