Ask what the player offers as a bribe then roll to see if it's enough. If the roll is more than the bribe, they don't take it (and may react poorly); if it's less they do. If it's exactly equal to the bribe they will take the money and shop the players later if it's profitable or convenient to do so.
To determine what to roll either pick a good sounding die or figure out the risk and level of corruption, picking the die between.
For instance, say they're bribing a guard to forget he saw them. You decide there's no risk that he'll get in trouble (d6) and that bribes often compliment his earnings (d12). Between those results is d8, so roll that. If there's a choice of two, choose the higher.
d6 - No chance of getting caught/Totally corrupt
d8 - Unlikely to get in trouble/gets bribed a lot
d12 - Slight risk/bribes compliment their earnings
d20 - Could get in serious trouble/not above a bribe now and again
d100 - Very risky/considers themselves to be honest
If the bribee is a shitbag you might be able to renegotiate. When the players pick another sum, take the old number and add a die of the next higher level. So in the previous example you'd add a d12 to whatever you rolled before.
Beggars aren't a symptom of inequality and economic pressure, they're thieves! Eyes and ears and whatever other limbs they still have of the Guild. For a bit of jink they might shed some light on the dark of things for a private operator.
If you give a beggar any money, roll d100 on the following table and add how much you gave them:
4-5 - The beggar coughs wetly. Save vs gross poor person diseases or you develop thick callouses on your face which cause people to be sickened by your presence.
6-7 - The beggar slips a few extra coins out of your purse with his sticky, sticky fingers. Lose d6 more without knowing it. Pretend you rolled a 16-50.
8-9 - He rants and raves, talking complete nonsense, pointing and waving his arms about. What at first seems to be a crazy person instead turns out to be a crazy wizard who still has some magic left from better times. Roll for a random spell across all spells you have to hand, or just flick to a random page of a convenient book. He casts that.
10-11 - He's not a beggar, but a noble young buck in disguise out to see how the hoi polloi treat the downtrodden. He is not impressed and challenges you to a duel.
12-13 - Thrilled at the attention he starts tell you his life story in excruciating detail. He'll follow you until physically restrained. He never stops talking and relates everything to something that happened to him in the past.
14-15 - The beggar will wink and give a knowing nod. He now has an unhealthy obsession with you and will turn up at weird and uncomfortable times, spotted through windows and down alley ways, standing over you while you sleep.
16-50 - Thanks you in his best beggar patter
51-60 - He'll drop an adventure hook on you. A fat tip
61-70 - He'll offer you a useful bit of mundane information.
71-80 - The beggar will answer one question you might have.
81-90 - Full access to any info that doesn't piss off the thieves' guild too much.
91-92 - The beggar will wink and give a knowing nod. Next time you find trouble in town you just might find a hobo with a hatchet coming to your rescue.
93-94 - He's not a beggar, but a noble young buck in disguise out to see how the hoi polloi treat the downtrodden. He's touched by your kindness and insists on returning it. He invites you to visit him, or offers you riches, or some other adventure worthy excursion.
95-96 - He tells you how he used to be an adventurer like you until he lost his legs/arm/nerve but he still has this map to a massive score. Wasn't interesting to the thieves guild 'cos they're thieves, not tomb robbers. Have it and good luck.
97-98 - The beggar gives you a piece of information so insanely obscure and useful that you couldn't possible fathom how he knew it. It could be an obscure spell, directions to physically enter heaven, the location of the bastard son of a king, anything. You never see him again.
99-100 - Gain a point of luck for being such a sweet guy. Spend this for a free success or to demand something goes your way.
101+ - Slow down big spender. The beggar freaks out, cries, and runs off. The thieves will hear about how much cash you carry and will plan a professional robbery or burglary in d3 days to clear you out.