Sybr green/safe/gold and blue light is a solution. You could buy this:

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or find a screen that will change the UV light to blue light (VWR has that for a few 100$) and with the right filter you d not even need to change your gel doc.

Another solution that works is simply to use the longer UV range (many transilluminator have a choice between long and short vawelength UV) and to work quickly. I have obtained sequences this way. I have seen special tips that are band shaped that allow a quick cut of the gel, so you can really turn the UV on for less than 1 minute, but just locating your band and then cutting UV off is good too.

they probably sent you the trace, I suggest you look at it, it should make the problem apparent. But to make short you have strong non specific fluorescence, and UV induced mutations can definitely be a cause for that. another way to solve your problem if you are sequencing a PCR, do not gel purify but tweak your PCR conditions to get only one band ( even at the cost of a lost of efficiency if possible, then just purify your PCR with a cleaning kit.

Another possibility, if your template is PCR product, is that your fragment is heterozygous with a deletion or insertion somewhere around the 100 bp point. Are you trying to sequence through an intron? Sequence up to that point will look good, but after the deletion or insertion, you will have two templates that are shifted out of phase with each other, and the traces will then look horrible. Follow canalon’s advice here – if the noise is just crap, you will see many signals of varying strengths at each site. If you have two signals out of phase, you should see only two main peaks of roughly equal height at each site. Having two peaks at each site would be less likely, although not impossible, if your template is plasmid – mutations do occur sometimes.