It's questionable as to how much real support they've got.
For example, in news reports the tv media will always give prominence to anyone who says positive things about the monarchy, but would not give hardly any prominence to Republican sentiments. The same applies to the newspapers.
Now, it may be that there is a genuine majority in favour of the monarchy (it would hardly be surprising given the amount of positive spin that they get), but I suspect it's nowhere near as much as the establishment would like to make you believe.
I always remember back in the mid 90's ITV did a programme on the monarchy asking people to ring in and vote about keeping them or not - surprise, surprise, there was a majority in favour. My experience however was that I tried ringing the anti monarchy number all the way through the evening and only managed to register a vote with about 5 minutes to go - I asked a number of friends if anyone else had tried ringing, and a few of my friends had - but in favour of the monarchy and they had got through straight away

Very surprising, bearing in mind that the majority of people ringing were ringing in favour
My friend commented at the time that I was naive in even thinking that there would be a genuine attempt by the media to get a genuinely fair result.
Anyway, my position is to retain my Republicanism and try to persuade others about the inequities of the monarchy, but at the end of the day, people have to make up their own minds on this issue, and hopefully many will see through the media's bias.