I was a member for almost the entire period of the club's existence, and during that time I managed to persuade a few of my friends to join including my cousin.
It was open to children aged 5-16, and I think it started in about 1979. It was initially called the Great Rail Club, and I joined in 1980 when I was 5. About a year after I joined it was renamed Rail Riders, possibly to coincide with the launch of BR's "This is the Age of the Train" advertising campaign featuring that disgraced DJ from Leeds (and little did we imagine what he got up to behind closed doors back then!).
I used to really like the "Rail Riders Express" club magazine, which you used to get about three times a year, usually once in the spring, once in the summer and once just before Christmas. In its early days you used to get a free gift with each issue, and it used to have a pen pals page where you could get in touch with other club members. I grew up in Putney, south-west London, and I had a pen pal who lived near Willesden Junction. I went to visit him a few times, taking the train from Putney to Richmond and then a Class 501 EMU on the North London Line to Willesden Junction. I think I first contacted him in 1983 or '84, and last saw him when my dad and I went with him to Reading to see the GWR 150th anniversary exhibition train in 1985.
In the late 1980s they brought out a new club logo, and redesigned the "Rail Riders Express" magazine. It got smaller but came out more often. Personally I preferred it in its original format.
Looking back on it now, I would guess that the club was aimed not necessarily specifically at enthusiasts, but generally at children who liked travelling by train, although it may have encouraged an interest in railways among those members who weren't already enthusiasts. Without wishing to sound sexist, railways have always been regarded as a largely male-dominated hobby, but Rail Riders seemed to have a good mix of members of both sexes.
Soon after I turned 16 in 1991 and reached the upper age limit, I learned that the club was closing down, presumably in anticipation of rail privatisation. A great pity, but I suppose it was a sign of the times.