Well, if you don't want to be in a tourist area, you should not look in South Beach. Try Key Biscayne.

Checkout ....
Sunny Isles Beach
Miami Beach North of 25th Street
Surfside
Bal Harbor

A beautiful sandy beach but not in a touristy area???? Define "touristy" and remember you are one. Unless you are going to lie on the beach in one of our state parks on the east coast you are gonna be hard pressed not to be somewhere there are tourists.

If you don't fashion the places the Beach Momma recommended (and since you are insisting on being within an hour of MIA that lets out wonderful possibilities such as Sanibel-Captiva on the west coast) you might look even further north at somewhere like Hollywood or even Fort Lauderdale. But you are going to see tourists and depending on the time of year lots of them. And don't forget, some of those beaches are close to a lot of very high rise buildings so try to look at a few pictures before deciding.

Yes, I think what you mean by touristy is important here:

hot dog stands?
rednecks and bikers?
old guys with toups and high waist pants?
all-night partiers and a big club scene?
topless German grandmothers?

Or ?

Apologies for my absence, I have been busy sorting out another part of our trip.
I understand we are tourists but what I mean is we are travelling with a one year old and would like some family beach time, nice food and relaxing break.
We do not want night clubs or the south beach vibe and don't want to be doing very much except enjoying the area.
We don't want more than an hour or so from the airport as we will have been doing a lot of travelling before this section of our trip and want an easy journey to and from the airport, but I am not saying we wouldn't go an extra 30mins if it were to make a big difference to our experience.
I will research the areas given. Thanks everyone.

If you can consider a (very easy) 90 minute drive to the airport, take a look at Islamorada in the Keys. I just returned from a few days there and will be singing its praises for a long time to come.

What is your price range per night?

Thanks will look into it.
Our budget would be roughly £100 per night so $1100-1200 for the week.
Obviously if we pay less we will have more to spend on enjoying ourselves so less is fine, more wouldn't really be possible.
Would love to be on or VERY close to a beach.

Take a look at hotels on Islamorada and Key Largo....if you take a look at the map you will see that these are narrow islands with water on both sides. Many hotels on both. I was astounded, really, by how lovely these places are.