WOW. There are lots of things you can do. One easy thing is to separate them by sexes and definitely do NOT cage all of them together. While yes, birds are very social and do like the colony setting, mixing males and females will more than likely result in breeding or hormonal behaviour that could lead to breeding. Do NOT let them inbreed (siblings, parents). It leads to a much higher chance of deformities and health issues.
1. Cage them separately. I covered this. Put the females in one cage, the males in another (make sure they're all spacious cages so none of the birds fight over territory - since you already have so many birds, I would suggest cages in this size range per gender: http://www.windycityparrot.com/Merchant2/merchant.mvc?Screen=PROD&Product_Code=19302242001&Category_Code=cagesbreed&Product_Count=16 since it allows for space to fly, space to stay away from others when they want their own space, space for toys, and space for perches), and the babies in another. When the babies' ceres are determinable for male and female, you can put them in with their gendered cage. Do this change AFTER all babies are fully weaned (about 8 weeks, but you should notice that they are eating on their own.)
2. Limit mating triggers.
(i) Daylight. Cover their cages with a dark sheet, blanket, or cage cover for at least 12 hours at night and give them total silence. If they continue to lay eggs, give them more and more dark hours. Worst case scenario, you could be working with 22 hours of darkness and only 2 hours of daylight, but these are the drastic measures of winter that make birds not want to breed.
(ii) No petting anywhere but the head. If you stimulate them by petting them, they'll want to nest because touching anywhere but the head is virtually telling them "it's mating time" because all other areas, specifically the back, under the wings, and tail/vent are areas that would be stimulated by a mate.
(iii) Monitor food. Give the birds only what they can eat and no more. If there's extra food, they'll think it's enough for babies and begin to try and breed.
(iv) Constant mental stimulation and change. Birds will not want to produce young in an environment that is moving too quickly or is too busy for the chicks to live in because it's dangerous. Give them many, many toys and many, many perches but keep them in rotation so that not one thing stays in one cage for more than a week. If the environment is always changing, not only are the adults getting mental stimulation, but it tells them that hatchlings will have trouble acclimating to the constant changes.
3. Make the eggs infertile. Some people have problems with this, but it's best you do it early rather than get stuck with more chicks that you cannot care for and that may die if not cared for properly by either you or the parents. When an entire clutch is laid (3 to 5 eggs, and birds lay eggs every other day), distract the parent(s) and take the eggs. Bring a small pot of water to a boil and boil the eggs for a couple of minutes. Dry them off, return them to the nest, and return parent(s). After 21 to 30 days, the parents should realize these eggs aren't going to hatch, lose interest, and you can throw the eggs away.
4. Remove the brooding box for sure, duh. Also remove any other places that the birds could hide and nest, whether it be a toy or the paper lining on the bottom of the cage (don't have to remove it, just put a grate or other sort of barrier in the way). Also, remove/put a barrier between all nesting materials such as paper so they won't feel the need to gather a nest.
5. Feed a balanced diet. This isn't necessarily about how you can stop the breeding, but because you have SO many babies, the parents are more than likely exhausted. Budgies need a diet of 80% high quality pellets such as Harrison's, Roudybush, Zupreem, or Totally Organic Pellets, 10% fresh fruits, vegetables, pastas, and grains (NO avocado, mushrooms, onion, tomato leaves, apple seeds, fruit seeds and pits, caffeinated or carbonated foods or drinks, salty or greasy foods, alcohol, or chocolate), and 10% seeds. Feeding the common diet of 100% seeds leads to malnutrition in the adults and chicks, obesity, and especially in budgies, fatty tumors. The females especially will need more calcium, so feed dark leafy greens often. The most popular I've noticed are Romaine lettuce, broccoli, bok choy, spinach, and kale. Feed spinach and kale in moderation because too much will be counterproductive.
6. When all birds are fully weaned, I find it very important to get all of them to see a certified avian vet ( http://www.aav.org/vet-lookup/ ) to make sure all are healthy and well. Otherwise, you could be setting yourself up for huge, major and much more expensive vet bills later if any of them get terribly sick. If one bird gets sick, you know the other 18 are definitely going to get sick because birds pass disease VERY easily and can hide the illness very well. However, a word of warning, be financially prepared for these visits. A typical avian vet visit for one bird is usually $50 to $75 USD - BUT this is much more reasonable than $200 bills per bird if they all get sick and need treatment.
If you know you can't handle them, it would definitely be wise to find other owners. Most people are looking for birds who come from good blood lines (pet store birds should never be bred because there is never much known about them, if anything, and they could have terrible genetic backgrounds and recessive traits that could hinder their chicks' survival chances, etc.), are healthy, and the vast majority of people want tame chicks. So you can definitely try to sell them, but responsible owners will make sure you know what you're talking about regarding your budgies and you should do the same when screening potential owners. Failing that, they should go to the humane society or an avian rescue where they will find good homes for them.
Good luck.