I don't know the area in question at all but we are in an area where we are rarely too far from strangles (although thankfully never had it ourselves). We are not far from a popular sales and lots of big livery yards allow people to pity buy and take them to the yard with no isolation procedure.
As such, we take reasonable precautions every time we take the horses off the yard. We only allow them to eat and drink what we take with us - no grazing at the showground and certainly won't use any communal water troughs.
We avoid our horses touching others or letting unknown people touch them. Likewise, we don't touch others unless we really have to (eg. in the right place at the right time to catch a loose horse) but wash hands after if possible). Obviously judges and stewards get close to them but they should not be touching horses anyway nowadays.
Wash hands after going to the toilet etc - lots of people will have touched the doors before you.
If a horse isn't 100% in itself then don't take it out - even if it isn't carrying any illnesses itseld, it's immune system may be more susceptible if exposed to anything whilst out.
If you see a horse that you think may be unwell, inform show organisers.
The good thing with strangles is it is only spread by contact (direct or indirect) unlike diseases such as foot and mouth which are airborn. This means that it is preventable if people are sensible and don't panic or at the other end of the scale try and brush an outbreak under the carpet.
So, in answer to your question, it is a completely personal decision to weigh up the risk in your area and what precautions you can reasonably take.
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