Great discussion. My own story....
My wife (V) has always been supportive of my blogging however it did take some time for her to make a bit of a mind shift from it being a hobby to it being a 'job'. I totally understand that - after all it is our future and that of our kids at stake.
My main advice with helping partners to make the transition from blogging being a hobby to a job (assuming that's your goal) is to:
1. take it slow - don't suddenly anounce that you're quitting your job and going to make a living from blogging
2. show them the potential - I remember making some graphs that showed my blogs earnings in the early days and projecting where the earnings would be in 6 and 12 and 24 months if the increases continued. Don't hype it up beyond what it is but show that you're on the track to it being more than a hobby (that is IF it is trending in the right direction).
3. if possible transition into blogging for an income - i was lucky in that I was working a number of part time jobs when I started blogging so it was possible for me to give them up one at a time as my blogging income grew. In this way we always had a backup income.
4. give them some control and be accountable - together V and I set a deadline at one point. When I was working a couple of days a week on my blog we set a 6 month deadline for it to become full time or else I'd have to go find a 'real job'. This was something we decided together - I think involving her in this showed that I respected her opinion but also made me accountable to her.
Ultimately keep in mind that the idea of making a living is foreign to many people so you need to bring them along for the ride. Also keep in mind that not everyone can make a full time living from their blog and that those that do take years to get there often - so don't rush your partner along.