mousikomisyo no zyuusyo no ran ha tiisai node, zi wo ookiku kakanai hou ga ii desu.

This is a chain of nouns, and each の links one noun to the next.

• 申込書 = application form
• 住所 = address
• 欄 = column / box / field / blank on a form

So:

• 申込書の住所 = the address on the application form
• 申込書の住所の欄 = the address field on the application form

In English, we often translate this chain backwards: the address field of the application form → more naturally, the address field on the application form.

Japanese often stacks nouns like this.

欄 means a space, box, field, or section for writing something, especially on a form or document.

So in this sentence, 住所の欄 means:

• the address field
• the box where you write your address

It is a common word in formal written Japanese, especially on forms:

• 氏名欄 = name field
• 住所欄 = address field
• 記入欄 = entry field / space to fill in

The は marks 申込書の住所の欄 as the topic of the sentence.

So the sentence is structured like:

• As for the address field on the application form, it is small, so ...

This does not mean 欄 is the subject in exactly the same way as English grammar. It means the speaker is setting up that part as the thing being talked about.

A natural English-style breakdown is:

• As for the address field on the application form, since it is small, ...

Because 小さい is an い-adjective, and before ので you use the adjective in its plain form:

• 小さいので = because it is small

You do not say 小さくので.

Compare:

• 高いので = because it is expensive/high
• 忙しいので = because I am busy
• 静かなので = because it is quiet
  
  • this one is different because 静か is a な-adjective

So:

• い-adjective + ので
• な-adjective + なので
• noun + なので

Both ので and から can mean because, but the nuance is a little different.

• ので sounds a bit more neutral, explanatory, and gentle
• から can sound a bit more direct or personal

In this sentence, ので fits well because the speaker is giving practical advice based on a reason:

• The address field is small, so it is better not to write too large.

This sounds soft and natural.

If you used から, the sentence would still be grammatical, but slightly more direct:

• 申込書の住所の欄は小さいから、字を大きく書かないほうがいいです。

Here, 大きく is the adverb form of 大きい, and it modifies 書かない.

• 大きい = big
• 大きく = big / largely / in a big way
• 書く = to write
• 大きく書く = to write big / write in large letters
• 大きく書かない = to not write big / not write in large letters

So:

• 字を大きく書かない = not write the characters too large

This is a very common pattern:

• 小さく書く = write small
• きれいに書く = write neatly
• 早く書く = write quickly

〜ないほうがいい is a very common pattern meaning:

• it is better not to ...
• you should not ...
• you’d better not ...

So:

• 書かないほうがいい = it is better not to write

The full idea is:

• 字を大きく書かないほうがいいです
• It is better not to write the letters too large.

This is usually advice, not a strict command.

Compare:

• 薬を飲んだほうがいいです = You should take the medicine.
• そんなに心配しないほうがいいです = You shouldn’t worry so much.
• ここで写真を撮らないほうがいいです = You’d better not take pictures here.

Literally, ほう means something like side or option.

So 書かないほうがいい is literally close to:

• the option of not writing is better

That is why it becomes:

• it is better not to write

This is an idiomatic pattern, so learners usually memorize it as a set expression:

• verb plain past + ほうがいい = should do ...
• verb plain negative + ほうがいい = should not do ... / better not do ...

Examples:

• 行ったほうがいい = you should go
• 行かないほうがいい = you should not go

字 can mean:

• a letter
• a character
• handwriting
• written characters

In this sentence, it does not have to mean just one single character. Japanese nouns usually do not mark singular vs. plural clearly, so 字 here can be understood more generally as:

• letters
• characters
• your writing

So 字を大きく書かないほうがいいです means something like:

• It’s better not to write the letters too large
• It’s better not to make your writing too big

Yes, that is possible, but it is a little different in feel.

• 字を大きく書かない focuses on the action of writing in a large way
• 大きい字を書く focuses more directly on large letters as the thing being written

So both can make sense:

• 字を大きく書かないほうがいいです
• 大きい字を書かないほうがいいです

The version in your sentence is very natural because Japanese often expresses this kind of idea with an adverb plus verb:

• 大きく書く
• 小さく書く
• はっきり書く

It sounds like advice, not a strict order.

That is because of 〜ないほうがいいです, which usually means:

• it would be better not to
• you probably shouldn’t

If the speaker wanted to sound stronger, they might use something like:

• 書かないでください = please do not write ...
• 書いてはいけません = you must not write ...
• 大きく書かないでください = please do not write too large

So your sentence is softer and more practical: Because the address field is small, it’s better not to write too large.

A useful breakdown is:

• 申込書の住所の欄は
  As for the address field on the application form,
• 小さいので、
  because it is small,
• 字を大きく書かないほうがいいです。
  it is better not to write the letters too large.

So the pattern is:

• [topic] は [reason] ので、[advice]
• As for X, because it is Y, it is better not to do Z

This is a very common practical sentence pattern in Japanese.