Vientiane's centre is small and flat, so the real difference between bases is how far you are from the Mekong and how much noise you want at night.
Mekong riverfront — the walkable core
The promenade side of town, five minutes from the night market, the temples and Talat Sao. You pay a premium for the river view and get the evening breeze and the sunset without a taxi.
Nam Phou and Ban Mixay — bars, bakeries and guesthouses
The block or two behind the fountain has the densest run of restaurants, bars and mid-range guesthouses. Everything is walkable, and it is the easiest base if you are only here a night or two.
Around Talat Sao and Lane Xang — cheapest and best connected
Next to the morning market and the central bus station, which is where the city buses to Buddha Park and the railway station leave from. Less atmospheric, noticeably cheaper, and unbeatable if you are moving on early.
Do not stay near the railway station
Vientiane Khamsavath station is about 14km out and there is nothing around it. Stay in the centre and take bus 28 or a taxi out on the day; allow 45 to 60 minutes plus the 30-minute check-in.