Paris is a compact city where most tourist neighborhoods are within 20 to 30 minutes of each other by metro, so there is no real "wrong" area to stay in. Each neighborhood does have a different atmosphere and practical trade-offs though, which can noticeably change the trip. It's worth choosing based on the kind of trip (a first, sightseeing-heavy visit, a romantic stay, a family trip, a tight budget) rather than following one universal recommendation.
Le Marais, central and lively
A few minutes' walk from Notre-Dame and the Centre Pompidou, with plenty of bars, shops and restaurants. Convenient for a short trip with minimal metro time, but also one of the more expensive and crowded neighborhoods, especially on weekends.
Saint-Germain-des-Prés and the Latin Quarter, the elegant, historic Left Bank
Literary cafés, the Jardin du Luxembourg, the Sorbonne, within walking distance of the Louvre and the Musée d'Orsay. A calmer, fairly upscale area, a good base for a museum-focused trip.
Montmartre, views and village charm but a bit out of the way
The Sacré-Cœur and its picturesque lanes are the main draw, but some streets around Pigalle are livelier at night (the area's adult entertainment district). Worth checking the exact address rather than trusting the name "Montmartre" alone.
Around the Eiffel Tower and the 7th arrondissement, quiet and residential
An upscale, family-friendly area, with tower views from some addresses, but further from nightlife and less directly connected to some other neighborhoods.
Canal Saint-Martin and the 10th/11th, young and popular with locals
Trendy bars and restaurants, good value for money, fifteen to twenty minutes by metro from the historic center. A neighborhood lived in by Parisians day to day rather than built around visitors.
Being near a metro station matters more than the neighborhood's name
Central Paris is small, but a fifteen-minute walk to the nearest metro gets tiring fast with luggage. Which line serves an address often matters more than which arrondissement it's in.
Tips
Favor a place near a direct metro line (lines 1 and 4, or an RER line, reach most major sights) to keep trips simple. Neighborhoods well outside the city's ring road are mostly worth it only on a very tight budget. Central neighborhoods like Le Marais or Saint-Germain cost more per square meter but save valuable travel time on a short trip. For a longer stay, slightly less central neighborhoods (the 11th, 13th, 19th) often offer much better value without seriously hurting access to the main sights.
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