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I can say **歩いて渡る** which translates to "to cross by walking". However, if I
would like to say "I am not going to cross by walking, but by some other
means", would I say **歩かないで渡る** or **歩いて渡らない**?

There are many other examples, most of which involve the use of the so-called
Japanese auxilary verbs that follow the te-form of the verb they "help" (ある,
いる, くれる, あげる, もらう, 行く, 来る, etc). I don't like this term because I view them as
just normal verbs.

Another example would be **食べている**. To say that "(I) am not eating", it would
be **食べていない** instead of the more logical structure **食べないでいる** , which would,
at least in my opinion, nicely and logically translate to "I am not in the
state of eating". This is because the verb that needs the negation is the verb
**食べる** , not the verb **いる**.

Explanations from linguistic points of view are also welcome.

* * *

I hate to treat this kind of verbs as auxiliary verbs. They are just verbs
chained together using serial verb construction, which exists in a number of
East Asian languages (Korean, Chinese, Thai, Vietnamese, etc), and at least in
my first language, I would treat them as normal verbs chained together to
indicate:

(1) a sequence of actions happening one after another

(2) a set of actions happening simultaneously as in 歩いて渡る and 食べている

(3) cause-effect

(4) method as in 歩いて渡る

So the concept is pretty much the same, except for the fact that in Japanese
these verbs are " **special** " when it comes to **negation**.
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grammar て-form negation linguistics subsidiary-verbs
Which verb receives a negation in a Japanese sentence?
It's

> [歩]{ある}かないで[渡]{わた}る  
>  cross without walking
>
> 歩いて渡らない  
>  not cross on foot

In this case you want the second option.

For "not eating" it is usually

> 食べていない  
>  I haven't eaten

whereas 食べないでいる is used to put emphasis on the duration of staying without
eating, but 食べていない also implies a continued state of being without food.

(There is more on this in [this
question](https://japanese.stackexchange.com/questions/3122/when-
is-v%E3%81%A6%E3%81%84%E3%82%8B-the-continuation-of-action-and-when-is-it-the-
continuation-of-state).)

The general rule seems to be to negate the auxiliary verb, unless you
particularly want the other meaning, e.g. compare

> しないであげる  
>  してあげない

where the first phrase means "doing something for someone by not doing
something else" and the second phrase means "not doing something for someone
by doing something else".

_Edit_. Since this answer has been downvoted and the OP still seems to be
active, let me just add what exactly I meant by "食べないでいる is hardly used" in a
previous revision of this answer.

The BCCWJ corpus yields the following results

```

     88 results 食べていない
    133 results 食べてない
      1 result  食べないでいる
    
```

so the form 食べないでいる needs the right context to be the natural choice. In case
of the corpus the sentence is

> 実際お菓子を **食べないでいる** ことは無理だと思います。  
>  I think that actually _staying without eating_ sweets is just not possible.
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In the Vて+V case, I think loosely translating て as "by" here helps give a
little intuition:

> 歩いて渡る "cross by walking"  
>  歩かないで渡る "cross (not by walking)"  
>  歩いて渡らない "not (cross by walking)"

However, this intuition does not hold with auxiliary verbs (補助動詞{ほじょどうし}), and
certainly not with inflectable particles (助動詞{じょどうし}).

With auxiliary verbs, you sometimes can't negate the base verb. For example,
the auxiliary verb いる has nothing to do with the verb 居る, it is just an
aspectual marker indicating the progressive in the case of action verbs, and
the stative form in the case of state-change verbs. And because of this, you
can only negate the entire thing, not the "inner verb", because there is no
inner and outer verb, there is only a single verb which is modified
grammatically.

With other auxiliary verbs, such as あげる, you can negate the inside because you
can "(not V) for someone" and also "not (V for someone)".

Essentially, some auxiliary verbs simply perform a grammatical function, and
with those, it does not make sense to negate the base verb.

With inflectable particles (ます, たい, etc.), they all only perform grammatical
functions, so it never makes sense to negate inside them and negation comes
last.
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