I live in new york city with my wife and two kids. We don't need a car
every day, but we need a minivan for shopping and weekend trips (car
will not be driven on weekdays).
I can afford approximately $7000, which can either buy me a 1999-2000
Toyota Sienna/Honda Odyssey with 150,000 miles on it, or a 2002-2003
Dodge Caravan with about 70,000 miles.

Which one would you reccomend?

I recently went through the same decision.  I ended up spending $6200
for an 2000 Mercury Villager with 70,000 (this was one year ago).  I
am very happy with the decision to go with a lower mileage vehicle,
rather than a higher mileage Toyota or Honda. Particularly given the
type of driving you have described I would go with the Dodge Caravan.

Below are three pages to read through that might help you make the decision.

1999 Toyota Sienna - Average Condition - $5700 - $7900
http://auto.consumerguide.com/Auto/Used/reviews/full/index.cfm/id/2318/act/usedcarreviewshowall/

1998 Honda Odyssey - Average Condition - $5700 - $6800
http://auto.consumerguide.com/Auto/Used/reviews/full/index.cfm/id/2174/act/usedcarreviewshowall/

2002 Dodge Caravan - Average Condition - $5900 - $7700
http://auto.consumerguide.com/Auto/Used/reviews/full/index.cfm/id/2491/act/usedcarreviewshowall/

Repair costs are comparable between them.  The Toyota seems to lead in
this department, but keep in mind that you are getting a vehicle that
is three years newer with the Dodge and most repairs should be farther
down the road.

The overall rating from the consumer guide website also favors the Dodge.

Personal brand preference is usually more of a factor than statistics
when making car purchases.  Buy what makes you happy! (but stick to
your budget)

The most important step in purchasing any used car is to have it
inspected by a mechanic that you trust.  Here are some guidelines for
what he should check:
http://autoadvice.about.com/od/buyingcarsused/qt/checklist.htm

If you require any additional information please let me know by
posting a request for clarification.

Thanks!

Dude, parking will kill you unless you aren't in Manhattan (and even
then). Why not join ZipCar or FlexCar?