In 1824 Ireland had a population of around 8 million so there are a lot of people with the same name. If the family came from County Down they may have been the Dempsters of Comber and surrounding district where Blakely and Beggs are also found. Unfortunately most Irish reacords up until about 1840 were lost in the 1922 fire in Dublin, including church records so finding a birth registration from 1824 (when registration was not compulsory) is difficult. Unless you have an idea where these people came from in ireland it will be difficult. You could start looing through the names in the International Genealogical Index (IGI) which any local branch of the LDS (Mormon) library will have for Ireland. You can look up the names in the surname index of <caora.net> and <freepages.genealogy.rootsweb.com/~rosdavies>
Most early records only listed the heads of households and virtually every early census has been destroyed so it is difficult to find children and wives. Dempster is the Manx word for a judge and common in County Down. If you can find James Dempster it will make your search much easier but you would have to be certain it was the right James Dempster.You could start looking for the James Dempsters in Griffith's Valuations of 1863-4 in Comber, Greyabbey, Drumbo, Drumgooland and Magheradrool, etc. at:
<members.cox.net/hayes1966/down>
Good luck with your search.