To say that I'm not the countries biggest royalist, would be a bit of an understatement, but neither have I seen any strong argument to abolish them.
They don't have a lot of power anymore (quite rightly in my view) and what powers they do have generally seem to be being phased out as we reform old laws. This bizarrely leaves them with very little purpose other than to be the Royal Family. As such, they're now more like a living cultural artifact or piece of artwork based which we like to show off to foreign nationals and they seem to relish it.
They are a big part of our history and hence our culture, but given this they should in some way reflect modern culture as well as the traditional.
It seems therefore sensible that as we are now in the 21st century these rules should be updated to include modern thoughts on equality and remove any issues regarding sex, religion, etc.