According to this question, the opening phase is not the best time to do so, because you get too few resources.
Harbors are good for trade, so I would imagine that a good time would be when one has "many" resources, but not necessarily the right ones.
So would that be late in the game, or even in the middle game? Or could game "circumstances," rather than game "phases," be the critical factor?
There are 2 types of Habour (or port) in the game, and they have different best times.
The 3:1 ports are always handy, more so in the late game. However, there are few reasons to have one early.
The 2:1 ports are worth building if, and only if, you have an excess of production in a particular resource. This may just develop late game (especially with brick and wood once you've got no expansion), or you may purposefully give yourself an excess with initial placement. This is most common with sheep.
However, rarely is it worth starting on a harbour unless it has excellent numbers (5-8, for example). In general, you should be able to build on a 3-resource hex and have an uncontested build to the harbour for your first settlement, which should give you more resources early on before the trade would be useful.
If a specific resource is exceptionally difficult to obtain (for example, brick has numbers 2-3-10) it could be worth your time investing heavily in a different resource and a matching port. A surefire return on investment would be getting three different numbers in the 4-10 range of the same resource (4-5-10 would qualify, but obviously 5-6-8 is better).
Having multiple good numbers on the same resource is a prerequisite here, because otherwise you'll be robber bait.
If you can meet these requirements with your starting settlements and take a port as well, go for it. Otherwise try to get settled near the desired port and head there later on.
Once you have this port, you wouldn't have to focus on building near specific resources. Instead, you can focus on building near different numbers, and fill the gap in your available resources with your port.
My harbor strategy is to place one or both of my original placements near harbors ... with the plans of making my first new settlement in a harbor -- preferably in a spot that borders the harbor and two hexes.
In my experience you particularly need a better trading ratio early in the game, when resources (both yours and others) are scarce, but I agree with those that suggest not placing either of your original settlements on a harbor.
A secondary benefit of putting your original placements near a harbor is that it prevents you from being boxed in, as it is usually easy to continue expanding around the edges of the board, settling on places that didn't look good for your original settlement, but are not at all bad later in the game.
Whether I use 3-1 or 2-1 harbors depends of course on which hexes I place my original settlements on, and whether I can reach the 2-1 that is most beneficial.
Some of my best games have been when I am heavily invested in a certain resource, and have the 2-1 port in that resource. (Because if I'm getting most of the wood, I can often get 1-1 trades with wood-deprived competitors first, then fall back on the 2-1 when necessary.)
Stating with an harbor is doable when for example one settlement is on a resource with at least 1 red number and you have the matching 2:1 harbor. You try to make a city as soon as possible to draw 2 cards at a time which with the trade basically gives you 1 resource of any kind.
Else, you can play with the harbor master tile expansion which gives 2 VP when you have 3 point worth of settlement on the shore (1 pt per settlement, 2 point per city). Game is won at 11 VP since it's easier to get 2 VP tile.
