I'm torn here, because Mom was obviously ill-prepared to fly with her sick toddler, but even if the kid was suffering from the stomach flu or something similar, he was already sick, so he was probably already past the point of passing on his illness to other passengers. Once you're symptomatic, you're generally not contagious anymore.
But the FAs were also rude in not helping the mother to clean up a little bit and make her time on the flight a bit more comfortable once everyone was secured and up in the air. It's no more and no less than they should do for any passenger.
Fortunately I've never been subjected to this kind of treatment by an FA while travelling. On New Year's 2000/2001, though, I flew to NY from CA to be with my long-distance BF at the time, and I was in the throes of a nasty flu that had developed a couple of days before Christmas and caused me to miss more than my intended 3 days of work. The first couple of days I was so bad off that I couldn't even croak a message to my boss to call in sick - I had to have my mom call me in sick because I couldn't talk. By some miracle the seat next to mine ended up empty, but I had my bottle of leaded Robitussin (Robitussin w/codeine) with me, a regular bottle of Robitussin, a bag of cough drops, and a box of Kleenex, which I threw away into a plastic shopping bag I'd brought with me just for that purpose, once they were used. So it was clear that I was sick, but I'd already been sick for a week and a half, so I was no longer contagious.
The FAs were nothing but wonderful to me and made sure I was comfortable on the flight, but every time I coughed, the elderly lady in the window seat (with an empty seat between us) glared daggers at me, until finally she leaned over and asked me, "Are you sick?" (what, the 2 bottles of medicine, the Kleenex and the cough drops weren't evidence enough?) I replied, "Yes, but I'm actually getting over it." (I actually didn't recover for another 2 weeks but that's beside the point).
She said nastily, "Well, if you're sick, you shouldn't be flying. There's a planeload of people you're contaminating, you know, and some of us are elderly and could get sick easily."
To which I replied, "Thanks for your concern, ma'am, but I've been sick for a week and a half, so I'm probably not even contagious anymore. Besides, if you had any qualms about sitting next to a sick person, this IS Southwest, so you had your pick of seats." I was the first in line, so I'd sat in the very first row, where I'd be near a bulkhead if I needed it.
Luckily, she got off at Kansas City, MO, the midpoint of the flight, but she spent the entire time grousing about me being sick and young people who were disrespectful to elders and so on and so forth, to the point where I wanted to give her one gigantic dose of leaded Robitussin so she would pass out and SHUT UP, but that leaded Robitussin was like gold to me, so I didn't (although I had another bottle stowed in my checked luggage). I couldn't help that I'd gotten sick just before my trip. Part of my trip was nonrefundable and I wasn't about to lose that money (because I couldn't afford to lay out more cash on another flight), and anyway I wasn't contagious anymore - once you're visibly sick, and you've been so for more than a week, there's no way you're contagious. When she got off the plane I made sure to give her a sweet smile and wish her a happy new year, but what I really wanted to do was cough on her for being so rude to me. For the whole 3-4 hours from CA to Kansas City. She just glared at me and stalked off the plane.
That was the last year I forgot to get my flu shot, although that hasn't always stopped me from being sick. Anyway, my whole point is, I wouldn't make assumptions about the mom - maybe her flight was nonrefundable and she didn't want to lose money on the trip and have to spend more money to take another flight. The OP's point was that the FAs were rude and unhelpful, and were I in that mom's position, I certainly would have reported the flight crew's unhelpful attitude. Sure, FAs are there to make sure FAA regulations are observed but the mom wasn't acting entitled - she was acting like a harried parent with a sick child. Illness doesn't choose convenient times to strike, and sometimes there's not a lot you can do, money or timing-wise, to juggle plans around, especially with air travel. People have jobs and lives to get back to, or sometimes can't afford to lose money on a nonrefundable ticket, or just can't change plans for whatever reason. One sick person on a plane is not an etiquette violation and is not the end of the world, unless that one sick person has radiation poisoning or something to that effect. Being a rude, unhelpful and unaccomodating airline employee, when part of your job is to ensure the comfort of your passengers, IS a violation, not only of etiquette but of your job responsibilities. I've never had a FA treat me with anything less than the utmost courtesy and attentiveness, so behavior like that would most definitely catch my attention and would cause me to think twice about giving my custom to that airline again, because the next person they could be that rude to could be me.