There are a few different places to take your dog in Port Melbourne. The first is the beach! There is a stretch that is leash free and the dogs love playing in the water together.
The 2nd is Lagoon Reserve just off Graham Street. Most nights there are a heap of dogs in the park so it's a lot of fun. On weekends generally in the morning there is sports so best to avoid those times, but overall a really great place to take your pooch.
The 3rd is Gasworks Park (technically in Albert Park) just east of Pickles Street. Lot's of bigger dogs in general here.
All 3 are great options for your dog in Port Melbourne.