What tools and tactics do countries employ to stop people from using spirits (typically under movement/concealment) from snuggling small valuable and illegal goods? More applicable for countries/corporations that have harder to cross borders.

Some tools that come to mind are:
1) Spirits and mages patrolling on astral
2) Goods picked up on the sensor net/radar

Truthfully?

Not a lot.

Given that a spirit could only carry (F+2/+4 as Str) x 10 kilo in goodies and they could be crossing just about anywhere the cost of magical patrolling/prevention on large swaths of territory would outweigh the value the cargo in most cases. Just remember that they have to be manifested for this so regular speeds apply and not astral zipping instantly anywhere.

Demonoe's Pizza for example uses Air spirits to deliver pizzas to most placesthroughout their area in good enough time that they are plenty hot and still dripping cheese, but if you wanted it to go to you while on stakeout in the Tir, you would be waiting a while.

Now as they could well fly under most radar if they know the defense setup, most sensors would pretty much be useless except in certain regions. Which is why security forces keep their exact methods a tight secret so they do not advertise whether their sensor fence suite on the border detects up to 10' off the ground or 25', but with some legwork and digging you could find this out.

Now if the KE in Seattle during bust discover that a certain mage is using spirit mules, then they may sic their magical departments to try and counter this specific person.

But they simply would not be able to afford the kind of coverage to shut down all such muling, despite their PR to the otherwise.

Spot patrols with watchers or astral mages -- they wouldn't see every spirit smuggling trip by any means, but the more that the method is used, the more chance they have of spotting it. Then find out where the spirits are delivering to, figure out who is picking up, start mapping out the chain that uses the spirits and target that.

Plus, most people who can summon spirits can make good money doing legitimate work, so the number who would get involved in something like this would presumably be pretty small, so it is never apt to be a huge issue. They'd like to stop it, but it is not apt to be their worst problem.

Seeker brought this up because it became an issue in our game last weekend. A player (not Seeker) wanted to bring along his Barrett sniper rifle for a little hunting trip in the Olympic peninsula but didn't want to deal with the border guards into Salish lands. So he figured he could summon a spirit to carry said gun over the border for him. I figured it would either be a case of no you can't do it all because astral security combined with physical would spot the spirit or it was something that happened all the time and was trivial.
Honestly, I think this is something the devs never looked and/or addressed at but it should be. Just how tight are astral borders? Spirits can patrol pretty big areas if you are talking buildings but the border around Seattle for example, is so large that it might as well not even be patrolled given how many spirits would be needed or how large the penalties would be to patrol if they used fewer spirits. Magic throws a real monkey wrench into the idea of tight borders.

The key drawback to this method is the time limit and the 'risks' of binding, if applied.

As a remote service, this is something fairly easy for a spirit to do - but in the case of an unbound spirit, they HAVE to arrive within the time limit of dawn or dusk (pretty sure that is still the rule in SR4] or they're going to basically drop that rifle in the middle of nowhere (or hide it someplace and let the mage know).

If that spirit also has the movement power, this becomes a little more plausible, granted.

A bound spirit can take their time, and depending on the spirit type doesn't necessarily have to have movement or be a high force. an air spirit can obviously flit about, but a beast or earth spirit probably has the extra strength/body to help cover the weight issue a bit without costing as much to bind - you still run that risk on the dice roll though, unless you count it as carefully done since you only really need 1 service to pull this off.

Otherwise, seems pretty doable with proper stealth and such!

I could see us making the mage in question accompany the spirit on the astral to inform the spirit open doors (maglock issue?) or to identify exactly what to bring bring back. Might be more lenient with spirit of man and some traditions which are more modern based (Chaos).

The block may be how would it access said item then the border.