I think it's pretty well established that if you cast Sanctuary on yourself, then initiate a grapple, sanctuary is broken because initiating a grapple is a "special melee attack." However, a situation came up that is a little less clear.

If a character has the War Caster feat, a spell focus that is accessible, one open hand, and the ability to cast Sanctuary, can she initiate a grapple with her action, then cast Sanctuary with her bonus action, and have the effect work on the grappled creature?

RAW, only the initiation of the grapple is an attack, after that, the grappled creature simply has the "grappled" condition, and may try and escape the grapple. The grappler in this case doesn't have to make additional special melee attacks to keep the target grappled.

In subsequent rounds, the grappler could do a number of actions (move, cast buffs or healing on the party, take the dodge action, etc) that wouldn't constitute an attack or spell that affects the enemy creature.

One aspect I'm not clear on is when the grappled creature tries to escape. The grappled creature has to succeed on a contested Acrobatics or Athletics against the grappler's Athletics. Would this satisfy the criteria of "mak(ing) an attack or cast(ing) a spell that affects an enemy creature"?

Does this work? Could the grappler maintain the grapple and the effect of Sanctuary in this scenario, as long as the grapple was initiated before the casting of Sanctuary?

Yes, this would work

Maintaining a grapple is not an attack or spell and does not deal damage

If the warded creature makes an attack, casts a spell that affects an enemy, or deals damage to another creature, this spell ends.

Maintaining a grapple falls under none of these categories, and neither do opposed ability checks in general. Thus, there is nothing RAW preventing this from working.

As long as you initiated the grapple before casting the spell, maintaining the grapple will not break the spell according to the rules.

Trying to break free does not trigger a Wisdom save

Escaping a Grapple. A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.

This attempt by the creature to escape also does not fit into the outline of things that will trigger the wisdom save effect of sanctuary:

[...] any creature who targets the warded creature with an attack or harmful spell must first make a Wisdom saving throw.

An opposed check is not an attack or harmful spell, so it would not count.