I certainly understand your trepidation on washing your lovely heirloom quilt. Cleaning your quilt does pose some risks because the colors might bleed, or the thread may break from the stress of the wet cloth. I have successfully hand washed several quilts that are even older than yours, and know it can be successfully done. Here are my recommendations:
1. Test for colorfastness on the backing fabric. Using a wet, white rag, and a small amount of a mild soap, gently scrub a small area of the backing fabric. If the color is set, it shouldn't bleed off onto your white rag. If you do get some of the blue bleeding onto the rag, the color is not set and the only type of cleaning I would recommend would be dry cleaning.
2. If the color doesn't bleed, you should be able to safely wash your quilt. Put three to four inches of lukewarm water in the tub along with a small amount of a gentle soap. They make soap specifically for cleaning quilts, which you should be able to find at your local quilt or fabric store. If that's not an option, look for a mild laundry soap at your supermarket. Put a large, flat sheet in the bottom of your bathtub, and lay your quilt onto that. The sheet will help distribute the weight, and take the stress off the seams, when you remove the quilt from the tub. Lay your quilt out onto the flat sheet, pushing it down into the water and let it get fully saturated, which might take a half hour or so, depending on the fabric content. Gently agitate the quilt with your hands, squeezing and pushing and pulling it around in the water, making sure to work your way around the quilt so you're swishing every bit, especially the edges which likely have trapped the most dirt. Then let the quilt soak for an hour or two. Agitate and let soak again. Soaking helps the dirt dissolve and work free. If the water is very dirty, drain it and put in fresh water and soap and repeat the agitation process. When you feel you've gotten out as much dirt as possible, rinse the quilt with lukewarm water. The easiest way is probably to use a bucket to dump clean, lukewarm water at the far end of the tub and let it drain away. Continue to do that until you feel the soap and dirt have been washed away. Then squeeze out the quilt as best you can, working from the far end toward the drain. Repeat that process until you've wrung out as much as possible, Then, grabbing the sheet underneath, gently lift the quilt out of the tub. It will probably still be dripping wet, so have a large plastic tub to set it into. Take your quilt outside and lay it on a dry, flat sheet spread out on the lawn. It's best to do this on a day that doesn't have full sun as sun causes fading and weakens the fabric and thread fibers. An overcast day with a breeze is perfect. The quilt will take most of the day to dry. Gently flip it over once or twice to make sure both sides get completely dry. You may even need to do this over the course of two days. If you don't have somewhere to lay your quilt outside to dry, you may want to put the quilt into your washer, after removing it from the tub, and use the gentle spin cycle to get out more water. Then lay it out on a flat sheet on the floor, or on a bed, and allow it to dry, flipping as necessary.
Here are links to a couple of publications on quilt care that might have helpful information:
http://www2.ca.uky.edu/hes/fcs/FACTSHTS/CT-MMB-065.pdf
http://www.extension.iastate.edu/sites/http://www.extension.iastate.edu/files/iowa/quiltcare.pdf