Whole-step vs half-step string crossing
There are two main types of string crossings when they occur in a stepwise motion: half-step or whole-step string crossing. Let's look at how they are different.
Whole-step string crossing
When you cross strings in a whole step, you don't have to open your hand. You can go to the next string with our normal perfect 4th hand frame.
Half-step string crossing
When a string crossing occurs in a half step, we need to open our hand by a half step to get to the next string. Though we should be used to stretching our hand by a half step, that does not mean it causes a strain on our hand.
Benefits of whole-step string crossings
Crossing strings in a whole step comes easier for us because it feels more natural to our hand. We can cross strings more easily, and it is easier to play in tune when we cross in a whole step.
Some examples
This is an excerpt from the 1st movement of "Symphonie Fantastique" by Berlioz.
By crossing strings in a whole step as in example 1, you are able to move more quickly to the next string than when you would in example 2. At a glance, 5th position might be a good solution, but then you'd have to deal with half-step string crossings. The 5th position causes another issue at the end of the 2nd measure where there are too many string crossings, but that's a topic for another time.
Let's look at another example. This excerpt is from the 1st movement of Mendelssohn Concerto.
You have to cross strings in a half step when you take the top fingerings. Compare that to the bottom fingerings where you stretch the 1st finger by a half step to play the E, hence avoiding the half step string crossing. You will find that the bottom fingering is much easier on your hand.
This next excerpt is from the 2nd movement of Violin Concerto by Brahms.
We have to change strings somewhere in the 3rd measure if you were playing the end of the measure on the G string. There are several ways to get to the G string, but let's look at the two that are shown in the example.
We switch strings between the 'A' and the 'G' in the top fingerings, which is a whole step. In the bottom fingerings, we switch strings between the 'B♭' and the 'A', which is a half step. When you try both fingerings, you will find that the top fingerings are easier on your left hand, and it is easier to play more in tune. And because it is easier on your hand, it is easier to be more expressive.
This next example is from the 2nd movement of Symphonie Fantastique by Berlioz.
Let's assume that we are in the 3rd position in the 3rd measure. If we continue playing in the 3rd position, we would go to the E string on the A, which is a half step string crossing. Instead, we can go to the E string on the G♯ if we apply the top fingerings. We have to extend the 1st finger just a little bit, but a whole step string crossing is worth considering.
A half step string crossing would be fine as well in this particular example. You already played the 'A' in the previous measure, so if you keep the 1st finger down you can do a smooth string crossing in spite of it being a half step string crossing. Some players might find it easier this way.
Think about each case
There is no one rule you can apply to everything when it comes to violin playing. You should learn rules but you should also think about each case whether applying the rule is appropriate.
During the thinking process, you will make many mistakes. You thought you came up with good fingerings, then a week later you discover better fingerings. Then a few years later, you realize your fingerings were not as good as you thought they were back then. If this happens to you, that means you are improving as a violinist. You now have a better understanding in fingerings. So think about them and make mistakes, and learn from them.