When making official measurements of lengths of coastline and nonlinear borders,
what precision is used for surveying all the zigs and zags?  If it were
infinitely precise, every coastline would be infinitely long, while lowering the
precision causes us to ignore inlets and peninsulas of various sizes.  Is there
a standard compromise?  What about changes due to tides?

I haven't found anything discussing a standard measure for coastlines.  
However, this page, which contains data about the length of the US Coastline, 
says that the measurements were made with a unit measure of 30 minutes 
latitude.  The same coastline length numbers appear in other places, including 
the CIA factbook, which suggests that they're fairly official.

http://www.infoplease.com/ipa/A0001801.html

Here's exactly what they say about the coastline measurements:

"Figures are lengths of general outline of seacoast. Measurements are made with 
unit measure of 30 minutes of latitude on charts as near scale of 1:1,200,000 
as possible. Coastline of bays and sounds is included to point where they 
narrow to width of unit measure, and distance across at such point is included."

30 minutes of latitude is half a degree, or about 55.6 kilometers.  Given the 
large scale of these measurements, the tide probably does not have a 
significant influence on the numbers.

That's exactly what I was looking for.

The answer is complete since you asked about the official method.

Neverthless, your question implies realization that scale used will
affect the measured quantity. If this in an independent realization,
you are
herby congratulated for your superb insight. The fundamental answer to
such
class of question is provided by concept of fractals:

http://www.laubender.de/fractaline/what_is_a_fractal.htm

Fractals as proper model for many natural phenomena
http://www.fortunecity.com/emachines/e11/86/mandel.html

were described by  polish born mathematician B. Mandelbrot 
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html
http://mathworld.wolfram.com/Fractal.html
and coastline was one of the examples he used.

 So, the comment summary is:

Coastline has fractal dimension (about 1.2-1.3) and when measured by
 fractal method 
http://hypertextbook.com/chaos/33.shtml

the result is independent of the scale.  Result describes how the
length
will change when scale (the size of measuring rod) changes.

 Briefly: result changes acording to a power law. 
 The exponent of the power is generalised dimension:

 The cube has power exponent of three, 
 square of two, 
 a smooth line one, 
 coastline about  1.28
etc
One day, officials too, will use this concept.