qǐng jìzhù zhīfùmìmǎ.

Here, 请 means please. It makes the sentence polite.

So 请记住支付密码 is a polite instruction or reminder.

A useful extra note: 请 can also mean to invite or to treat someone, but not in this sentence. Here it is just a politeness marker.

Mandarin often leaves out the subject when it is obvious from context.

In an app, on a website, or in a notice, the hidden subject is usually 你. So this sentence is understood as:

请你记住支付密码。

The shorter version sounds very natural in instructions and interface text.

Yes, absolutely.

• 请记住支付密码。 = shorter, common in written instructions
• 请你记住支付密码。 = also correct, a little more explicit

Both are natural. In things like banking apps and warnings, the shorter form is especially common.

记住 is more natural here because it means remember and keep it in mind or commit it to memory.

In 记住:

• 记 = remember, note down, memorize
• 住 = a result complement

So 住 does not mean to live here. It shows that the action reaches a result: the information is successfully retained.

That is why 请记住支付密码 sounds like Please make sure you remember the payment password.

They are related, but not the same.

• 记住 focuses on successfully keeping something in your memory
• 记得 often means remember in the sense of not forget or have a memory of

Examples:

• 请记住这个号码。
  Please remember this number.

• 我记得他的名字。
  I remember his name.

• 记得带钥匙。
  Remember to bring your keys.

In your sentence, 记住 is better because it is telling someone to retain the password clearly in memory.

Because 支付密码 is a noun compound, and Mandarin very often forms compounds without 的.

Here:

• 支付 = payment, to pay
• 密码 = password, passcode, PIN, code

Together, 支付密码 means payment password or payment PIN/passcode.

Chinese often puts the modifier directly before the noun in this kind of fixed expression. Similar examples are:

• 登录密码 = login password
• 银行卡 = bank card
• 手机号码 = mobile phone number

Using 支付的密码 here would sound unnatural.

By itself, 支付 is often a verb meaning to pay.

But in 支付密码, it is functioning as a modifier inside a compound noun. In other words, it means the password used for payment.

This is very common in Mandarin: a word that is verbal in one context can modify a noun in another context.

密码 is a broad word. Depending on context, it can mean:

• password
• passcode
• PIN
• code

So 支付密码 could be translated in different ways depending on the system:

• payment password
• payment PIN
• payment passcode

If the system uses numbers only, English might prefer PIN. If it uses letters, numbers, or symbols, English might prefer password or passcode.

The pinyin is:

qǐng jìzhù zhīfù mìmǎ

Tone by tone:

• 请 qǐng = 3rd tone
• 记 jì = 4th tone
• 住 zhù = 4th tone
• 支 zhī = 1st tone
• 付 fù = 4th tone
• 密 mì = 4th tone
• 码 mǎ = 3rd tone

A natural rhythm is:

请记住 / 支付密码。

One pronunciation note: 请 keeps its 3rd tone, but in normal speech it may sound a bit shorter and lower than an exaggerated textbook 3rd tone.

Yes, that is correct.

• 请记住支付密码。 = Please remember the payment password.
• 请记住你的支付密码。 = Please remember your payment password.

Adding 你的 makes the ownership explicit. Leaving it out is very common when the context already makes it obvious whose password is meant.

In app messages and warnings, Chinese often prefers the shorter version.