There's no definitive in-game/canonical evidence as to which brother is older.
It's obvious that they both care about each other, however the Wiki entries for Sans and Papyrus just list them as 'brothers', and everything in-game pointing one of them being the older sibling is circumstantial at best - most likely played this way so as to leave it up to the player's own interpretation/head-canon.
Some parts of the story points to Sans being younger:
- Papyrus encourages & cajoles Sans for his laziness:
IT'S BEEN EIGHT DAYS AND YOU STILL HAVEN'T... RECALIBRATED. YOUR. PUZZLES! ... YOU GET LAZIER AND LAZIER EVERY DAY!!!
- Sans admires his brother
my brother's a real star....
- Who encouraged him to take a job as a sentry:
...he's the person who pushed me to get this sentry job.
Other pieces of the story point to Sans being older:
- Sans has a 'real' job & takes care of his brother/pays the bills
- Sans reads Papyrus bedtime stories and gives him presents
Sans cares for and admires his brother. He reads him bedtime stories and gives him presents under the identity of Santa.
- Undertale Wiki
There are some other fan-theories & explanations out there, for example, that they are as old as the creation dates of the fonts the characters are based on: Comic Sans and Papyrus, but again, as these are just fan-theories and aren't confirmed in-game in any way.
Finally, it's worth mentioning that the developer of the game - Toby Fox - has also previously tweeted that Papyrus might use the Japanese word 'Aniki' (i.e. 'Older Brother' or 'Superior' in Japanese):
Papyrus would definitely use "aniki." i think. what's your take.
- Toby Fox. September 10, 2015 (Internet Archive)
Again, this isn't covered in-game and Toby has since deleted this statement. Thus it shouldn't be interpreted as 'Word of God' (TV Tropes). Even if the statement hadn't been recanted, there is still enough evidence to point either way in game, thus it would still be open to interpretation.
Until (and if) we get another source of official canon material (such as a second game, comic, movie etc) that builds upon or embellishes the story, we don't have the evidence to say which brother is older than the other. It's ultimately up to you to decide.