I believe it really depends on your set up, your program, your mare and your foal(s) AND what shows you are looking at going to.
When we arrived in NC, we brought our bred mares with us - from Montana. Then we acquired more and continued to breed them. These mares were the ponies our 3 daughters and I were riding - often daily. We trail rode, did lessons, used them for lessons for others and showed - with foals as young as 1 week of age right beside the mare.
The mares were used to the foals being handled right from day one, they were used to foals being 1 - loose running with them; 2 - tied up or held outside of the round pen or arena while "mama" was working or 3 - being stalled with a same year playmate in either a pen or in a barn or if actually at a show, in the trailer. The foals penned up had access to creep feed buckets, hay and water and mare's milk replacer. They often seemed to enjoy this time that they spent w/o their mom's and once they got used to it - everyone was happy.
While we did haul to shows from 1997 to now - they were open or 4h shows - 1 day type affairs and usually less then 250 ENTRIES (not horses. Horse/handler/rider often entered much more than 1 class) w/i 2 hours' drive from us. Several per year were on the premises of the barn we leased our 5 acres from for 7 years. The barn and arena were 2 miles from our pasture - the mares often got ridden up to the barn/arena and the foals were led by my husband or i or a friend. Our oldest daughter was able to "pony" babies off of most of the ponies/horses she was riding. OR they were loaded into the trailer with the other young ponies and allowed to nurse when the "mama" made it to the showgrounds. Those 7 years the fewest # of ponies we had at a show was 4 and one of those was just a week old. Sometimes we'd have as many as 10 ponies tied around our 4 horse (14' long) stock trailer!! That week old foal, due to the scorching heat, spent part of the show day in the cab of the truck in front of the air conditioner and would hop out of the cab - surprising anyone close by. Depending on number and which show we were at, we'd be parked in the back and sometimes the foals could be turned loose to lay down while their moms stood tied. This was just on "our" show grounds.
On other show grounds we didn't turn the foals loose unless we were right there (such as eating or changing clothes or resting) and then they had a well fitted halter and proper lead rope on. That made it easy to "police them up" if necessary. A couple of times, we did plan ahead and hauled mares w/ foals to the showgrounds the day before - the mare and foal were stalled together and when the two mares were shown the foals were put in the same stall. The two foals were also shown in foal/weanling halter classes - that was 2001 and the foals were both born in April and the show was in June so they were 2 1/2 months old (one was 1/2 Arab, the other pure Arab - both by the same Arab stallion).
The mares' were kept well vaccinated (w/ the hurricanes that swept thru in 98/99/2000, they often got EWT 4x yr dependent on mosquito populations in our area), the foals received vaccinations as soon as old enough and also kept up on boosters. While they were led around the showgrounds, they weren't "allowed" to nuzzle at other horses and were usually kept quite separate. We used our own hay/water and feed buckets - not a communal one (it now drives me crazy to go to some of the Draft Horse Events and the only water available is from a huge stock tank that every one drinks from! I've learned to carry my own in 5 gallon, lidded buckets). But yes, spectators and other competitors that came to "ooo and aaawwweee" over our ponies were allowed to pet/handle them (could be bad). Hauling those ponies to shows and other events are part of what helped us sell them - often to other show competitors that then continued to show/promote them but sometimes not.
It never occurred to us that it was wrong to take foals to shows until I got involved with several breed organizations here in NC and especially with Shetlands and Minis. We showed mares and foals in the 60s thru the 80s when I was growing up - at H/J shows, breed shows, overnight camping/trail rides and rodeos. Not all of our babies went (we didn't have nearly as many then as we have had since getting into the ponies) nor did every mare get bred - but they were used to being handled and moved away from each other. We culled based on how the mares/foals did - from working under saddle in a "job", how easily they got and stayed in foal, easy to foal out and all the way to how they did when a foal at foot. A mare that was ugly about being handled after she foaled LEFT our small place in a hurry. We simply couldn't, and didn't need to, deal with that type of attitude.
For us, now, Shetland and Mini shows are a minimum of 1 state away (well a couple are here in NC sometimes). I don't know if I would haul current year foals now or not - but part of that is simply logistics. The girls are no longer at home. The young granddaughters don't have the exposure to the ponies that their mom's had and they are scared and have a tendency to totally ignore what they are told and I spend more time dealing with that. Makes it really hard to deal w/ mares and foals or indeed ANY ponies at a show venue - especially now that I'm working so many more hours than I used to and the ponies aren't handled anywhere near the way they were at one time.