Copenhagen is one of the more expensive cities in Europe to date in, and the easy way to spend money here is to pretend that's the only option. It isn't. Some of the best dates I've been on in this city cost less than a single round at a fancy cocktail bar. Here are the formats that hold up, with rough prices so you can plan.
The harbor bus and a hot dog (around 60 kroner)
The 991 and 992 harbor buses are part of the regular public transport system, which means a normal city ticket (around 24 kroner for two zones) gets you on. Take it from Nyhavn to Refshaleøen, then back. You're on the water for forty minutes total, you see the Opera, the Royal Library, and the Inderhavnsbroen bridge, and the views are better than the harbor cruise that costs 100 kroner more. Pair it with a pølse from DØP near Rundetårn (around 35 kroner for an organic hot dog) and you've got a 60-kroner date that doesn't feel cheap.
Swimming at Sandkaj or Islands Brygge (free)
From roughly May through September the harbor baths are open and free. Sandkaj in Nordhavn is the one I'd pick for a date because it's calmer than Islands Brygge, the cafes nearby are good (Gorilla for coffee, Aamanns for smørrebrød), and the crowd is local. Bring towels, bring a swimsuit, swim for an hour, sit on the dock with an ice cream from Hansens Is (around 40 kroner). Total cost: maybe 100 kroner for two if you're frugal.
Islands Brygge is the more famous version and works equally well, just busier. Either way the swim itself is free and the water is clean enough that the city tests it daily.
A library date at the Black Diamond (free)
The Royal Library extension on the harbor, known as the Black Diamond, is open to the public and free to wander. The atrium is enormous, the views over the harbor are some of the best in the city, and the cafe serves coffee at normal prices. Spend an hour walking through, sit in the reading room, get coffee in the lobby cafe (around 40 kroner each). It's a real date format, not a backup. Pair it with a walk along the canal afterward.
Torvehallerne grazing dinner (around 200 kroner per person)
Torvehallerne, the food market at Israels Plads, gets called touristy, but it's where a lot of locals do their weekend shopping. As a date, the trick is to graze instead of eating at one stall. A smørrebrød at Hallernes (around 70 kroner), a glass of wine at Unika (around 60 kroner), an oyster or two at Fiskerikajen (around 25 kroner each). Two people can eat well for 400 kroner total. It's interactive in a way a sit-down dinner isn't, and the noise of the market gives the conversation cover.
A long bike ride to the beach (transit cost only)
Copenhageners take the bike ride itself as the date. Rent bikes if you don't have them (Donkey Republic is around 100 kroner for a few hours) and ride to Amager Strandpark, the city beach. It's about twenty minutes from the center. Once you're there, the beach is free, the dunes are good for sitting, and the sea is swimmable in summer. Bring beer and bread from a Netto on the way. Total cost for two: around 250 kroner including bikes and supplies.
The alternative ride is north along the coast toward Bellevue Beach in Klampenborg, which is longer (about an hour) but takes you past Hellerup and the harbor. The train back is around 35 kroner if you give up on biking the return.
A bottle of wine at the lakes (around 120 kroner)
Danish public drinking laws are relaxed, and a bottle of decent natural wine from a shop like Rødder & Vin or Vester Vinhandel runs around 100 to 150 kroner. Take it to the lakes (Sortedams, Peblinge, Sankt Jørgens) on a summer evening, find a bench, drink it slowly. Bring two glasses, not plastic cups. Total cost for two: under 200 kroner. This is the actual local move and people do it constantly from May through September.
Free museum days
Several museums in Copenhagen are free or have free days. The National Museum on Ny Vestergade is always free for the permanent collection. The Glyptotek is free on Tuesdays. Statens Museum for Kunst is free for the main collection. Plan around these and a museum date costs you a coffee in the cafe, nothing more.
A practical note on prices
A pint in a normal Copenhagen bar is around 60 kroner. A glass of wine is 70 to 90. A coffee is 35 to 45. A reasonable dinner per person is 200 to 350. If you know those baseline numbers, you can build a date in your head without surprises. The expensive Copenhagen and the affordable Copenhagen are the same city. You just have to pick which one you're in tonight.