The short answer is an unqualified no
Currants belong on the short list of ordinary kitchen foods that are truly dangerous to dogs. The small dried currants used in baking are dried grapes — the dried fruit of specific grape varieties — and grapes in every form, fresh, dried as raisins, or baked into food, have caused sudden kidney failure in dogs. This is not a rich-food-will-upset-his-stomach warning; it is a poison warning, and currants should be handled in a dog's home the way you would handle any other toxic substance: sealed, labeled, and out of reach.
Why no amount counts as safe
The frustrating and frightening feature of grape-family poisoning is its unpredictability. Some dogs have eaten grapes without visible effect while others have become seriously ill after a small handful, and veterinary science cannot yet predict which dog is vulnerable or identify a safe threshold below which nothing happens. The damage, when it comes, targets the kidneys, the organs a dog cannot do without. Because there is no reliable safe dose, the only defensible rule is zero: no currants as treats, no currant-studded baking shared, no exceptions for a dog who stole one once and seemed fine.
Where currants hide in a household
Currants rarely sit in a bowl like grapes do; they hide inside other foods. Fruit loaves, hot cross buns, Christmas-style puddings and cakes, mincemeat, scones, granola, and trail mix can all carry currants alongside raisins and sultanas, which carry the same risk. A dog begging for a corner of your fruit loaf is begging for exactly the ingredient that must never reach him. Store baking currants in a sealed container on a high shelf, keep baked goods containing them off low tables and out of lunchbox reach, and make sure children understand that these particular sweet things are never shareable.
If your dog eats currants, act immediately
Do not wait for symptoms, because by the time a dog shows signs — vomiting, unusual tiredness, loss of appetite, drinking or urinating strangely — the kidneys may already be under strain. Contact your veterinarian or an animal poison service the moment you discover the theft, even if it is the middle of the night, and be ready to say how many currants or how much baked food disappeared and when. Follow the clinic's instructions exactly, which will usually mean being seen straight away. Quick action gives a dog the best possible chance, and no clinic will fault you for calling about a genuinely dangerous food.
One naming confusion is worth clearing up, because it changes nothing about the rule. Garden blackcurrants are a different fruit from baking currants, but labels are inconsistent, mixed dried-fruit products blend several fruits together, and no one at a kitchen counter can reliably sort one tiny dark dried fruit from another. Given what is at stake, the safe household policy treats every product sold as currants — and every mixed fruit product that might contain them — as forbidden to the dog, full stop.
Put currants on the written never-feed list
Poison rules cannot rely on everyone in the house simply knowing. Guests, children, and new sitters all meet your dog without your knowledge of grape-family risks, and a written warning travels where your voice cannot. List currants, grapes, raisins, and sultanas together on your dog's never-feed list in KinsLe, and Care Teams displays that warning to every person assigned to feed or treat your dog, which closes the gap a holiday fruit cake could slip through.