Did they have a reasonable expectation that you would be the primary "bread-winner" of the family. Do you have children and is the spouse concerned about their well-being ? Did you look at worst-case financial situations for your family and discuss that with your spouse ? Do you have a deadline for making reasonable money through your startup and have you told your spouse that you will go back to a regular job if the startup fails ? Does your spouse think that a startup may reduce your family-time ? Some of these questions may be relevant, some may not and there are other questions that I didn't list.
Ultimately, though I'd go back what I first said. Your post doesn't say why your spouse/partner/SO is unsupportive. I think you need to find out why he/she is unsupportive before asking the next question of how to "convince them of the value of their endeavour"