If you're hosting a birthday celebration dinner, it's your responsibility to pay the cost. In the event it's a large party (many guests) and some of those guests who can afford to chip in offer to do so, arrangements should be made for them to either give you money toward the cost ahead of time or reimburse you after the party, but NOT at the time the bill is paid or even at the restaurant so their involvement is noticeable to anyone else. This will prevent embarrassment to those family members who cannot afford to share the cost.
If those who can afford to pay something do not make the offer until the bill is presented, say something such as, "Thanks...I'll handle this with my credit card." Later, you may thank them for their offer and give them an opportunity to follow through on their offer. The key is to keep this "transaction" separate from the party so that no one who is unable to do the same will be offended.
If a guest or guests at the party feel compelled to offer paying something, but you know they can't afford it (and you invited them), put them at ease by saying, "Hosting the party is my gift to (name of honoree)." This graciously lets them feel they don't need to insist on doing something that may put them in a financial bind.
When you host a party at a restaurant, it should be considered no less your responsibility for expenses than if you cooked and served the meal in your own home. However, if you've been doing this with extended family for three decades, there's probably some understanding among the more affluent members of the group of everyone's general financial standing. This makes it easier to accept offers of help on the expense of the dinner party from those who you know can afford it.