How does the relationship between the Capulet and Montague families impact the servants in Romeo and Juliet?

The servants are involved in the feud because it affects the honor of their families.

Shakespeare begins the play by demonstrating to us that even the lowest members of the Montague and Capulet families are affected by the feud.  In the first scene, two Capulet servants, Sampson and Gregory, are ready to fight servants of the Montague house just because they are Capulets and those men are Montagues.

SAMPSON

I strike quickly, being moved.

GREGORY

...

The servants are involved in the feud because it affects the honor of their families.

Shakespeare begins the play by demonstrating to us that even the lowest members of the Montague and Capulet families are affected by the feud.  In the first scene, two Capulet servants, Sampson and Gregory, are ready to fight servants of the Montague house just because they are Capulets and those men are Montagues.

SAMPSON

I strike quickly, being moved.

GREGORY

But thou art not quickly moved to strike.

SAMPSON

A dog of the house of Montague moves me.

GREGORY

To move is to stir; and to be valiant is to stand:
therefore, if thou art moved, thou runn'st away. (Act 1, Scene 1)

Abraham and Balthasar of the Montague clan are the dogs that move Sampson.  The infamous thumbing of the nose insult causes a swordfight to break out between the two groups.  Soon Benvolio tries to break up the fight, but Tybalt makes it worse.  Even the Lords Montague and Capulet get involved.

CAPULET

My sword, I say! Old Montague is come,
And flourishes his blade in spite of me.

Enter MONTAGUE and LADY MONTAGUE

MONTAGUE

Thou villain Capulet,--Hold me not, let me go. (Act 1, Scene 1)

The prince is so annoyed that the family feud is spewing into the streets that he makes a proclamation punishing anyone who gets involved with death.  He leaves scolding Lord Capulet and telling Montague he will lecture him later.

The feud between the Montagues and Capulets is so bad that Tybalt gets angry enough to start a fight when he sees Romeo, Lord Montague’s son, at his cousin Juliet Capulet’s ball.  Lord Capulet tells him to leave off because a fight would cause a stir among the guests.  However, you can be sure he would not be happy to see his daughter making eyes at a young Montague.

A servant would feel himself honor-bound in those days to defend his house.  This is why the servants of the house of Montague are so ready to fight the servants of the house of Capulet.  They insult each other and spoil for a fight because they feel just as involved in the feud as actual family members.