Do I have to replace both Airport base station and modem? I have an Airport base station and Network DG814 Modem Gateway both bought in 2002 and use over the years for wireless access at home to a family's worth macs and one work windows laptop.  Demon is our provider. 

Although everything is working we are resetting the modem a few times a week and speeds seem a little slow. I expect the modem needs a replacement. I looked at Airport Express and Airport Extreme options, but they need the modem too. We don't need multiple networks or back up, or anything fancy. 

Is there a reliable solution ?
As you mention Demon I am going to assume you are in the UK and therefore more likely looking at an ADSL connection, not Cable or other types etc.

Almost any ADSL Modem Router from Belkin/DLINK/Netgear etc will do the trick nicely, they should all be compatible with your Mac.

However, there are a few scenarios where continuing to use a (new) Airport Extreme could be of benefit depending on your usage and requirements.  First, the new Airports (to be fair, I believe others are starting to offer this also) will allow you to run up to 3 seperate wireless networks from a single device.  This may seem overkill, but for my set it's quite useful:

1) I run a Wireless N network operating at 5Ghz for all the stuff in the house that supports it, iMac, Macbook, iPad and Apple TV (I think) - this is faster and suffers less interference than...
2) A Wireless G network operating at 2.4Ghz for everything else that doesn't understand Wireless N and/or 5Ghz (PLaystation, Wii, Phones, PSP, work laptop, etc etc)
3) A "guest" network that has no WEp/WPA security, but is ringfenced from the other 2 to allow things my my sons Nintendo DS to connect easily (as it only supports WEP, which is next to useless) as well as let freinds use it when they visit.

To make the actual internet connection you can get a standalone ADSL modem, but they are quite rare, you would likely have to order online to get one, they won't stock them in PC World etc.  I actually use a full Dlink ADSL Modem Router which I turn off the wireless, and put into "bridge mode" to allow the Airport to control the connection.

Having the external "point of access" to be the Airport instead of a different Modem Router allows things like Back to My Mac to work far more easily without having to play around with port forwarding etc, which is another reason why I do it, also if your internet connection goes down for any reason and you need to reboot it etc, then it does not affect any of your local LAN traffic, i.e. Apple TV streaming, iTunes Homeshare, etc.