I read that the stars in the center of globular clusters are incredibly close to one another and that these stars can be spaced 100 to 1000 times closer than the stars near the sun. Because these stars are so closely spaced, would their heliospheres interact with each other? There are binary stars whose coronas interact and create powerful x-rays. Would something like that happen with those central globular cluster stars, but on a much larger scale, what with the larger amount of stars involved?
Similar to what @Barrycenter said in his comment, There's a wide range of distances between 100 or 1,000 times closer on average and as close as binary stars.
Alpha Centauri A and B get as close as 11.2 AU. The Sun and Alpha Centauri are roughly 270,000 AU apart so at 1,000 times closer, the close stars in a globular closer would still be (ballpark), 250-300 AU from their nearest neighbors. In a crowded ball of stars you're not likely to get binary systems that far apart, so most of the interaction that you're looking for would be in close binary systems. Neighboring stars in a globular cluster would mostly be too far apart to have much interaction.
There's also a world of difference between the corona and the heliosphere. The Sun's corona extends about 12 solar radii or about .05 AU. The heliosphere, while, it's one, not a sphere and two, probably has some variation in it's boundary, extends roughly 100 AU (121 by this estimate), over 2,000 times further than the corona.
Binary systems like Alpha Centauri A & B probably have a shared heliosphere and/or some form of heliopause between them where their two solar winds meet. It would be interesting to study that in more detail. I'm not sure how much is known about the heliopause between two binary stars, especially given that they orbit each other. I'm not sure what form it would take - perhaps that would make a good question.
Binary stars need to be much closer than 11.2 AU to have corona to corona interaction. Stars that close are probably quite rare but they do happen. Here's one example of a binary star system so close they're actually thought to be touching.