Pencil grasp questions are common with parents so I will try and provide some thoughts on the matter.   There are lots of components that go into handwriting competence (see my previous blog on handwriting ), but the most obvious one is grasp.   

How do grasps develop?  Children learn to grasp objects from the time that they shake a rattle.  Development begins at the core (proximal) and develops outward (distally).  So, if you don't have good core you may have difficulty with handwriting. Efficient handwriting uses the small, intrinsic muscles of the hand.  These intrinsic muscles are designed for dexterity, not strength.  The muscles on the side of your thumb are your movers and the ones near your pinky finger are your stabilizers.  A proper grasp ensures that handwriting is being produced by these small muscles so that there is a precise between movement and stabilization.  

How are our toys changing grasps?  Kids learn proper grasp patterns by playing with fine motor toys, finger foods, and coloring.  This develops and refines over time.  However, as toys change, so do fine motor exposure.  Small pieces in toys have long since been replace with big pieces due to choking hazards.  Thus, a reduction in exposure to small toys/parts requiring dexterity.

What does a proper / functional  grasp look like?  A functional grasp is one that allows the little muscles of the hand to work, not the big one.  There are 3 optimal grasps: Dynamic Tripod, Modified Quadrupod, and the Adapted Tripod Grasp. 

What does an inefficient grasp look like? A child should start developing an efficient grasp by their fifth birthday or within that year.   Most inefficient grasps are developed, not only from lack of proper fine motor toys/experiences, but also from writing /drawing with large or heavy markers and pencils, along with just writing too early (before the hand has fully developed).  This creates habits which can be hard to break.  Here are some pics of INEFFICIENT GRASPS:

How can I improve my child's grasp?
Around the age of 3, I recommend using crayons that have been broken down to no more than 1" long.  If your child still tends to put everything in the mouth, then you'll want to either supervise or wait until they have kicked the habit.  For girls, DO NOT, I say, DO NOT break them in front of them as their heart may break!   For boys, they will enjoy it and will likely help you in breaking the rest of the crayons in the house.  :--)  

Activities that strengthen grasp are:
1) clothespins
2) spray bottles
3)picking up coins and putting into piggy bank
4) hiding and seeking beads/coins in theraputty or in PlayDough
5)  stringing beads (start with large and go smaller as child improves)
6) craft projects with scissors, stampers, hole punchers, etc
7)  allow child the time to get dressed fully by him/herself
8) if you need more ideas, go Google crazy! There are tons of great posts by O.T.s and teacher

My child doesn't have a good grasp, but the handwriting is fine. Should I worry?  
If your child is past 1st grade them changing grasp is quite difficult.  Having an inefficient grasp will not make your child lose vocational ability.  For the most part, an inefficient grasp may cause issues as the demands of amount and speed of handwriting increase (think middle school). Your child may complain of : hand fatigue, arm fatigue, not keeping up with note taking, and sometimes a callous on the middle or ring finger.  Cursive may be more difficult to execute because it requires more precision and flow.   If the student is past 1st grade, I simply show him/her some options for a grasp (such as the adapted tripod below) and explain why and what they can do when their hand hurts from writing.