I like it a lot; it's a shame it's been used so often for pets, but it doesn't give me pet vibes as much as say, Bella or Daisy do. I think I'd be more likely to use it for a middle name.
I don't think I'd go out of my way to use a nn for Jasmine; Jazz or Jazzy might come naturally at least at times, but jasmine's so pretty and simple enough that it doesn't need a special nn.
Jasmine Adriana
Jasmine Alexis
Jasmine Brielle
Jasmine Charlene
Jasmine Christina
Jasmine Diana
Jasmine Dominique
Jasmine Elaine
Jasmine Faye
Jasmine Gloria
Jasmine Irene
Jasmine Katrina
Jasmine Kimberly
Jasmine Leilani
Jasmine Leigh
Jasmine Noelle
Jasmine Paige
Jasmine Rae
Jasmine Raquel
Jasmine Sabrina
Jasmine Valentina
Jasmine Yvonne
Before you criticize a man, walk a mile in his shoes. That way, when you criticize him, you're a mile away and you have his shoes!
Steve Martin