This answer is based on a question migrated from here to Japanese Language and Usage a while back.
Below I've pasted the old answer in its entirety, but I want to address some things in the newly-worded question. Your concern with Japanese students' spelling lies primarily in the fact that they are writing in romaji, and not in English. You have to drill into their heads this idea that romaji does not mean English. No English speaker who doesn't understand Japanese sytems of romanization would look at "geisya" and know to pronounce "sha," and that should be your pedagogical basis. As someone who teaches Japanese students English I cannot emphasize enough that romaji does not mean English. That said, I think you can teach them the modified Hepburn style, though you don't need to call it Hepburn. Just say "し" sounds like "shi" in English, so that's how we write it. "しゃ sounds like 'sha.'" Make sure they understand that "sya" as romanization (that is, kunrei-shiki) is formed by し (s) + ゃ (ya), and that because this romanization is based on kana orthography it will not be understood in English.
In short, to summarize those rules, you'll be using Hepburn without macrons or elongated vowel sounds. Some exceptions apply (as detailed below), but if they're at the level where they're just learning how to spell I think we can just tell them to learn the exceptions later or as they come across them in regular use. I think one of the bigger issues you'll find Japanese students have is that they can't even spell non-Japanese English words correctly and will try to write based on romaji anyway. That's a whole other can of worms, though.
Here is my answer from JLU in full:
Generally speaking in English words will be spelled in a way that resembles Hepburn romanization but without macrons or any special characters, instead just writing it 'plain.' There are not that many words in regular use in English that are of Japanese origin, but those that are do follow patterns (with a few exceptions).
For example, we say Tokyo, Kyoto, and Osaka, with no macrons and no doubling of vowels (No Toukyou, Kyouto, or Oosaka). The other features of Hepburn are present, though. So for example しゅ will be "shu" (not syu), as in Kyushu. じゃ will be ja, as in ninja (cf. ninjya). しゃ is "sha" as in shamisen, し will be shi, as in sushi. つ will be tsu, as in tsunami, and づ "dzu" as in kudzu. ふ will be fu, as in futon. There are many other words that follow the patterns as expected and that don't approach any touchy areas of romanization.
There are some exceptions, though. For example 能 (theater) is spelled as "Noh," which breaks from the patterns and uses the h to add emphasis to the longness of the o sound. Shiitake maintains the double i. Sometimes ん is written using an "m" instead of "n" as in tempura. You also have 浮世絵 which gets hyphenated, as in ukiyo-e.
There are some other words that are from Japanese but underwent complete spelling changes rather than pure romanization, like 大君 (tycoon) and 力車 (rickshaw).
So yeah, the correct English spellings usually follow Hepburn pretty closely, but there are exceptions. As forms of romanization, Kyusyu and the like aren't wrong, but if we're writing in English then I would say those should be corrected to writings that are understandable to English speakers, so definitely Kyushu if it's English.
If you want to peruse a list, you can take a look at the Wikipedia article List of English words of Japanese origin.
Also, just to quickly address a comment that snailplane made about place names, the rule applies the same ways but with occasional exceptions. For example we write "Niigata" instead of "Nigata," and in fact I see even the smallest, most obscurely-named villages get macron-free Hepburn romanization on road signs and stuff.
Another very small thing to note is that we don't usually use an apostrophe to denote the division between "んい", as in Kenichi/Ken'ichi (けんいち, with the former being the more common).