Hi chickens,
I am a dog owner and a dog trainer. So hopefully I can offer some advice...
Firstly, does dog get fed at the table? If you or your kids are giving in to his charms then why would dog stop doing something that is working? Human food is very appealing to dogs, dogs see your family unit as a pack, they want to eat what you are eating to be part of the pack. Plus human food normally smells very appetising compared to the dry pellets they get! If you haven't already, cease all under the table feeding. Tell your kids how important this is too. My dog went through a begging phase recently, I couldn't work out why until I caught my 7yo nephew, feeding him under the table!
Secondly, your dog doesn't like stair gates, and he is not alone, I have never met a dog who isn't bothered by them! So I am not surprised that he is crying, if you give him ANY reaction (comforting him or even telling him off) he will know that crying works to get your attention. So you must completely ignore his attempts at contact if you do decide to continue with the stair gate.
Alternatively a great solution would be to have him wait patiently and quietly in the kitchen but away from the table. So here are some things to try... Your dog needs a place where he can feel safe and happy to wait. Presumably he has a dog bed that he sleeps on and is happy to sleep on when the family is watching TV etc. He needs to know the command "bed" for this to work. So hopefully he already knows this one, if not, reply to the post and I will provide more details. He also needs to know the command "stay".
Put his bed in an area in the kitchen where it is away from the table. Ideally put the bed in a place where it can remain there all the time. If you don't normally keep dogs bed in the kitchen, it might be worth investing in another bed. So that whichever room you are in, there is a designated dog area.
Now to meal time... don't feed your dog before you. The last in the pack gets the last pickings. So wait until everyone at the table has eaten and cleared away plates before dog gets a look in. When you start cooking put your dog on his bed or out of the room with the rest of the pack. When you are ready to serve up the food, use the "bed" command and then the "stay" command. Your ideally dog needs to understand "stay" and have release word so that he doesn't come to the table. If you don't have a release code, every time your dog gets up from his bed to go to the table, stand up and lead the dog back to the bed with a firm "bed" command and then a "stay" and then go back to eating. Be firm and keep at it. It is not going to happen overnight and your dog will take a while to adjust to this new structure but stick to it and don't give in! Puppies are easier to teach but you CAN teach an old dog new tricks!
I hope this helps, if there are still issues you need to look at the wider picture and what other behaviour he exhibits and what this is a result of. You may wish to consider getting professional help, if so I recommend looking on the KCAI list for a credited instructor in your area.
Good luck!