It's really just trial and error. You can start doing the veteran versions right at VR1 (it's called Veteran Rank for a reason).
Just make sure the dungeon scales to the lowest character in your group (i.e. you should be group leader, if you're the lowest character). Also, if you're good at healing, there's nothing speaking against you trying the dungeons at higher ranks. Just keep your group alive and avoid damage at all costs (impossible in some dungeons).
Only exception would most likely be the veteran version of City of Ash, since that dungeon is harder than the others.
But basically, if you have no issues running the normal version, imagine the veteran version to be a beefed up variation with new/changed enemies. So try to get good at the normal ones, then take on veteran versions.
As a start, Wayrest Sewers is probably the easiest veteran dungeon (the second boss is a healing/damage mitigation challeng; the others are piece of cake), followed by Spindleclutch (first two bosses are DPS race/healing challenges).
TL; DR: Go and grab a group that is okay with you trying them for the first time, then just do it.
Remember that most people/groups are very helpful and newbie friendly, if you tell them. Be honest, tell them you're new, and you shouldn't have any problem, even if it means the group needs a bit longer to clear something.
