The repertoire has a seemingly bewildering array of string-crossing patterns compounded by the mixing of slurred and separate notes. For problematic passages it is worth using video or a mirror to help analyse the pattern in detail. For an example of detailed analysis see Hodgson, p.95, for his discussion of separate triplets across two strings.
There are two basic ways of crossing strings: one is by using the whole arm working as a single unit, and the other is by using a pivot action, with either the forearm or hand.
The whole-arm cross is our default string-cross action and is usually taught first. Since the arm works as one unit all the variables of hair inclination, weight of arm and leverage are approximately the same on each string.
Follow the curve of the bridge and cross as smoothly as possible. Keep the same action going from the E to the G-string as you do going from G to the E-string.
When crossing over three or more strings the whole-arm crossing is commonly used, since a pivot action causes too much change in bow hair inclination (see Example from Bach Partita).
The whole-arm action has an instinctive natural direction, with the down-bow falling from G to E-string and up-bow rising from E to G-string (see Whole-Arm with Bowstroke). To feel the difference, try reversing this to play up-bow slurred from G to E-string.
Pivot string crossing
This is used when crossing repeatedly between two strings (see Example from Bach Partita) or when crossing strings at a fast tempo for just a few notes. The upper arm does not cross to the new string but either stays behind or moves to a mid-point between the two strings while the forearm or hand take the bow across.
To maximize resonance and minimize string-crossing noise, concentrate on the horizontal drawing of the bow across the string, keeping the vertical action, that is the string-crossing action, to a minimum. Imagine that the screw of the bow describes an oval shape, not a circular one.
Forearm pivot action
This has the advantage of allowing greater bow length and more possibility to vary the length of stroke to support the phrase.
Hand pivot action
This has the advantage of a smaller action, for fast small crossings.
Direction of the pivot string cross
It can feel easier crossing with the down-bow going to higher strings. However a fluid bow technique will allow the string cross to work in both directions and will give you more options. Often we use a mixture of the two actions. (See Score of Allegro assai - violin & viola, from Sonata No.3 in C Major by Bach).
Place in the bow
Monitor your place in the bow. When you play at the point you will need a larger string-crossing action than when playing in the middle.
Three bows per note
Practise passages of string crossing using three bows per note. This allows the bow time to settle on each string (to check arm levels and point of contact) while maintaining the correct pattern of up and down-bows.
Example from Bach Partita
Whole-Arm with Bowstroke
Tone Production - Point of Contact
Planning and Practice - Three Bows Per Note
Scores - Allegro assai - violin & viola
Resources - Bibliography
Hodgsons book Motion Study and Violin Bowing includes diagrams of the path traced by the bow arm, which he called cyclegraphs.