Hayling Island is safe. The currents follow the shore so you won't get washed out to sea and there is a dedicated swim area with buoys marking out a route of about 1 mile with a pub right next door. The water is shallow so you can easily put your feet down if you are finding it too much.
Portsmouth is nice too, especially up the east end of the sea front where there are less people. There is also a designated swim area here, but the currents tend to be a lot stronger. Again, they run parallel with the shore but sometimes they ccan be very hard to swim against. There is a fantastic cafe that does the best fry ups and cake ever, called the 10th hole.
If you want somewhere scenic then there is the Hamble at Manor Farm Country Park, just off the M27 near Botley. It's tidal so you get the experience of salt water but it tends to be more sheltered from the wind. Check out the tides first though as there is no water at low tide. You are best sticking to an hour either side of high tide.
You can also swim at Calshot, but again this is best around high tide as the current is very strong at other times. Best to avoid this one on the weekends when the weather is good as it is very popular and you will find it hard to park.
I also know people who swim at Milford-on-Sea and Bournemouth / Boscome. These have sandy beaches so are easier on your feet when getting in and out of the water but I do find that they are more exposed to the wind than the other spots so the water can be very choppy.